Box Product of Mackey Functors\\ in Terms of Modules
Zhulin Li
The box product of Mackey functors has been studied extensively in Lewis's notes. As shown in Thevenaz and Webb's paper, a Mackey functor may be identified with a module over a certain algebra, called the Mackey algebra. We aim at describing the box product, in the sense of Mackey algebra modules. For a cyclic $p$-group $G$, we recover a result from Mazur's thesis. We generalize it to a general finite group $G$ in this article.
1509.07051.tex
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\title{Box Product of Mackey Functors\\ in Terms of Modules}
\author{Zhulin Li}
\date{\today}

\begin{document}

\begin{abstract}
The box product of Mackey functors has been studied extensively in Lewis's notes.
As shown in Thevenaz and Webb's paper, a Mackey functor may be identified with a module over a certain algebra, called the Mackey algebra.
We aim at describing the box product, in the sense of Mackey algebra modules.
For a cyclic $p$-group $G$, we recover a result from Mazur's thesis.
We generalize it to a general finite group $G$ in this article.
\end{abstract}

\maketitle
\section*{Introduction}
A Mackey functor is an algebraic structure, related to many natural constructions from finite groups, such as group cohomology and the algebraic K-theory of group rings.
The study of Mackey functor in abstract began in 1980s.
Dress\cite{Dress} and Green\cite{Green} first gave the axiomatic formulation of Mackey functors.
Several equivalent descriptions of Mackey functors were given by Dress\cite{Dress}, Lindner\cite{Lindner}, Lewis\cite{Lewis} and Thevenaz\cite{Thevenaz}.
Specifically, Lewis\cite{Lewis} introduces box product and Thevenaz\cite{Thevenaz} describes a Mackey functor as a module over the Mackey algebra $\mu_R(G)$.

In this article, we give an inductive description of the box product of two left $\mu_R(G)$-modules.
The main goal is to construct this box product explicitly and to prove that it is equivalent with the box product of two Mackey functors.
When $G$ is a cyclic $p$-group, we recover the formula of Mazur\cite{Mazur}.

\section*{Acknowledgement}
This work was made possible in part by support from 2015 Spring MIT Undergraduate Research Opportunities Program (UROP).
I would like to thank my supervisor Dr. Emanuele Dotto at MIT for proposing this interesting topic and for his comments that greatly improved the manuscript.
I also wish to thank Dr. Zhiyi Huang at HKU for helpful discussions, which provide lots of insight.

\tableofcontents
\nomenclature{$G$}{a finite group}
\nomenclature{$R$}{a unital and commmutative ring}
\nomenclature{$\mu_R(G)$}{the Mackey algebra}
\nomenclature{$e$}{the trivial subgroup of $G$ with only one element}
\nomenclature{$\pi^H_K$}{a map from $G/K$ to $G/H$, mapping $gK$ to $gH$}
\nomenclature{$c_{g,H}$}{a map from $G/H$ to $G/^gH$, mapping $kH$ to $(kg^{-1})^gH$}
\nomenclature{$^gH$}{$gHg^{-1}$}
\nomenclature{$H^g$}{$g^{-1}Hg$}
\nomenclature{$K'< H$}{$K'$ is a proper subgroup of $H$}
\nomenclature{$K\leq H$}{$K$ is a subgroup of $H$}
\nomenclature{$\omega(G)$}{an intermediate category of definition for Mackey functors}
\nomenclature{$\Omega_R(G)$}{a category of definition for Mackey functors}
\nomenclature{$B^G$}{the Burnside ring Mackey functor}
\nomenclature{$r_f$}{restriction of $f$}
\nomenclature{$t_f$}{transfer of $f$}
\nomenclature{\A}{the category of Mackey functors}
\nomenclature{\B}{the category of left $\mu_R(G)$-modules}
\nomenclature{$C_{g,H}$}{$G/^gH\leftarrow G/H=G/H$ or its value under some Mackey functor}
\nomenclature{$R_K^H$}{$G/K=G/K\to G/H$ or its value under some Mackey functor}
\nomenclature{$I^H_K$}{$G/H\leftarrow G/K=G/K$ or its value under some Mackey functor}
\nomenclature{$[J\backslash H/K]$}{the set of representatives of $J\backslash H/K$}
\printnomenclature

\section{Mackey functors}
Throughout this article, we assume that $G$ is a finite group and that $R$ is a unital, commutative ring.
There are several equivalent definitions of Mackey functors and we concentrate on two of them here.
Before giving the definitions, we introduce an auxiliary category $\Omega_R(G)$ and the Mackey algebra $\mu_R(G)$.
\subsection{The category $\Omega_R(G)$}
We recall the definition of the category $\Omega_R(G)$ from \cite{Lewis}.
A finite group $G$ gives rise to a category $\omega(G)$ whose objects are finite $G$-sets and where the morphisms from $X$ to $ Y$ are the equivalence classes of diagrams of $G$-sets $X\leftarrow V\to Y$. Two such diagrams are said to be equivalent if there is a commutative diagram
\begin{center}
\begin{tikzcd}
 & V \arrow{dd}[swap]{\sigma}\arrow{ld}\arrow{rd} &\\
 X & & Y\\
 & V'\arrow{lu}\arrow{ru} &
\end{tikzcd}
\end{center}
where $\sigma$ is an isomorphism of $G$-sets.
To define the composition of morphisms, we consider a morphism from $X$ to $Y$ represented by a diagram $X\leftarrow V\to Y$ and a morphism from $Y$ to $Z$ represented by a diagram $Y\leftarrow W\to Z$.
We form the pullback
\begin{center}
\begin{tikzcd}
&&U\arrow{ld}{f'}\arrow{rd}{g'}&&\\
& V\arrow{rd}{f}\arrow{ld}&&W\arrow{ld}{g}\arrow{rd}&\\
X&&Y&&Z
\end{tikzcd}
\end{center}
which defines a diagram $X\leftarrow U\to Z$, hence a morphism from $X$ to $Z$.
Such a pullback always exists and we can express it explicitly as \[U = \{(v, w)\in V\times W:f(v) = g(w)\},\] where $f'(v, w) = v$ and $g'(v, w) = w$.
By defining addition in $\hom_{\omega(G)}(X,Y)$ as
$$(X\xleftarrow{\alpha}V\xrightarrow{\beta}Y) + (X\xleftarrow{\alpha'}V'\xrightarrow{\beta'}Y) := (X\xleftarrow{\alpha+\alpha'}V\sqcup V'\xrightarrow{\beta+\beta'}Y),$$
$\hom_{\omega(G)(X,Y)}$ becomes a free abelian monoid, as shown in \cite{Thevenaz}.
Extending the scalars to the ring $R$, we get a free $R$-module $$\hom_{\Omega_R(G)}(X,Y) := R\hom_{\omega(G)}(X,Y)$$ on the same basis as that of free monoid $\hom_{\omega(G)}(X,Y)$.
Let $\Omega_R(G)$ be a category with the same objects as $\omega(G)$ and hom-set $\hom_{\Omega_R(G)}(X,Y)$ defined as above.
$\Omega_R(G)$ is an $R$-additive category.

\subsection{The Mackey algebra $\mu_R(G)$}
We define the Mackey algebra as $$\mu_R(G) := \bigoplus_{H,K\leq G}\hom_{\Omega_R(G)}(G/H, G/K),$$ where the multiplication is defined on the components in the direct sum by composition of morphisms in the category $\Omega_R(G)$, or zero if two morphisms cannot be composed.

For convenience, we recall the notation from \cite{Shulman}.
\begin{defn}
Let $f:X\to Y$ be a $G$-equivariant map of finite $G$-sets.
Then two spans
\begin{center}
\begin{tikzcd}
& X\arrow{rd}{f}\arrow[-, double equal sign distance]{ld} &\\
X & & Y
\end{tikzcd}
\text{ and }
\begin{tikzcd}
& X\arrow{ld}{f}\arrow[-, double equal sign distance]{rd} &\\
Y & & X
\end{tikzcd}
\end{center}
are called the restriction $r_f$ and transfer $t_f$, respectively, of $f$.
\end{defn}
\begin{defn}
Let $K\leq H\leq G$ and $g\in G$.
Then define $R^H_K$, $I^H_K$ and $C_{g, H}$ as
\begin{equation*}\begin{split}
&R^H_K = r_{\pi^H_K}: \left(G/K=G/K\xrightarrow[]{\pi^H_K} G/H\right)\\
&I^H_K = t_{\pi^H_K}: \left(G/H\xleftarrow[]{\pi^H_K} G/K=G/K\right)\\
&C_{g, H} = t_{c_{g,H}} = r_{c_{g^{-1}, ^gH}}:\left(G/^gH\xleftarrow[]{c_{g,H}} G/H = G/H\right),
\end{split}\end{equation*}
where $\pi_K^H:G/K\to G/H$ denotes the canonical quotient map, mapping $gK$ to $gH$, and $c_{g, H}:G/H\to G/^gH$ denotes the conjugation map, mapping $kH$ to $(kg^{-1})^gH$.
Observe that $R^H_K, I^H_K$ and $C_{g, H}$ are all elements in Mackey algebra $\mu_R(G)$.
\end{defn}

For conjuation, the notation $C_g$ is preferred to $C_{g, H}$ for simplicity when there is no ambiguity.
It is easy to check that the following identities hold:
\begin{enumerate}[(1)]
\setcounter{enumi}{-1}
\item $R^H_H = I^H_H = C_{h,H}$ for all $H\leq G$ and $h\in H$
\item $R^K_J R^H_K = R^H_J$ for all subgroups $J\leq K\leq H$
\item $I^H_K I^K_J = I^H_J$ for all subgroups $J\leq K\leq H$
\item $C_g C_h = C_{gh}$ for all $g, h\in G$
\item $C_g R^H_K = R^{^gH}_{^gK}C_g$ for all subgroups $K\leq H$ and $g\in G$
\item $C_gI_K^H = I_{^gK}^{^gH}C_g$ for all subgroups $K\leq H$ and $g\in G$
\item $R^H_J I_K^H = \sum_{g\in [J\backslash H/K]} I_{^gK\cap J}^J C_g R^K_{K\cap J^g}$ for all subgroups $J,K\leq H$
\item $\sum_{H\leq G}I_H^H$ serves as the unit in $\mu_R(G)$.
\end{enumerate}

We use $[J\backslash H/K]$ to denote the set of representatives of the double coset $J\backslash H/K$.
In fact, the structure of Mackey algebra $\mu_R(G)$ is rather simple:
\begin{lem}[from \cite{Thevenaz}]
Hom-set $\hom_{\Omega_R(G)}(G/K, G/H)$ is a free $R$-module, with basis represented by the diagrams
\begin{equation*}\begin{split}
I^K_{^gL}C_{g,L}R^H_L = \left(G/K\xleftarrow[]{\pi^K_{^gL}c_{g,L}}G/L\xrightarrow[]{\pi^H_L}G/H\right)
\end{split}\end{equation*}
where $g\in [K\backslash G/H]$ and $L$ is a subgroup of $H\cap K^g$ taken up to $H\cap K^g$-conjugation.
\end{lem}
In other words, the Mackey algebra $\mu_R(G)$ is generated by $R^H_K$, $I^H_K$ and $C_{g, H}$'s as an $R$-algebra.

\subsection{Definition of Mackey functors}
\begin{defn}[from \cite{Dress}]
A Mackey functor is a $R$-additive functor $M:\Omega_R(G)^{\text{op}}\to R\text{-mod}$.
They form a category with natural transformations as morphisms and we denote this category by \A.
\end{defn}

It is shown in \cite{Thevenaz} that the category $\B$ of left $\mu_R(G)$-modules is equivalent to $\A$ via the following equivalence of categories
\begin{equation*}\begin{split}
\Phi:\A &\longleftrightarrow \B\\
M&\longmapsto\oplus_{H\leq G}M(G/H)\\
\left(G/H\mapsto I_H^H N\right)&\longmapsfrom N.
\end{split}\end{equation*}
Since $\sum_{H\leq G}I^H_H$ is the unit in $\mu_R(G)$, a left $\mu_R(G)$-module $N$ can be graded into $N=\oplus_{H\leq G}I^H_HN$.
Observe that the multiplication by $R^H_K$ maps $I^H_H N$ to $I^K_K N$ and the other grades to zero.
Similarly, multiplication by $I^H_K$ maps $I^K_KN$ to $I^H_H N$ and the other grades to zero.
Multiplication by $C_{g,H}$ maps $I^H_H N$ to $I^{^gH}_{^gH}N$ and the other grades to zero.
Also note that $I^H_H\Phi M = M(G/H)$ for later use.

\section{The Box product in \A}
The box product is a symmetric monoidal structure on \A.
The box product in $\A$ has been studied in \cite{Lewis} and we summarize it in this section.
The result we are more interested in is that maps from the box product of $M$ and $N$ to $P$ can be characterized by Dress parings.

Given two Mackey functors $M,N\in\A$, we can form the exterior product
\begin{equation*}\begin{split}
M\overline{\square}N:\Omega_R(G)^{\text{op}}\times\Omega_R(G)^{\text{op}}&\longrightarrow R\text{-mod}\\
(X, Y)&\longmapsto M(X)\otimes N(Y).
\end{split}\end{equation*}
\begin{defn}[Box product in \A]
\label{def:box_product_1}
The box product $M\square N$ is defined to be the left Kan extension of $M\overline{\square}N$ along the Cartesian product functor $\times:\Omega_R(G)^{\text{op}}\times\Omega_R(G)^{\text{op}}\longrightarrow \Omega_R(G)^{\text{op}}$.
\begin{center}
\begin{tikzcd}
\Omega_R(G)^{\text{op}}\times\Omega_R(G)^{\text{op}}\arrow{r}{M\overline{\square}N}\arrow{d}{\times} & R\text{-mod}\\
\Omega_R(G)^{\text{op}}\arrow[dashrightarrow]{ru}[swap]{M\square N}&
\end{tikzcd}
\end{center}
\end{defn}

If a Mackey functor $M\in\A$ is implicit, we use $r_f$ for both a morphism $\left(X=X\xrightarrow[]{f}Y\right)$ in $\Omega_R(G)$ and its value $M(Y)\to M(X)$ under $M$.
Similarly for $t_f$.

\begin{lem}[from \cite{Lewis}]
\label{lem:dress}
A map $\theta:M\square N\to P$ determines and is determined by a collection of $R$-module homomorphisms $$\theta_{X}:M(X)\otimes N(X)\to P(X)$$ for every finite $G$-set $X$, such that the following three diagrams commute for each $G$-equivariant map $f:X\to Y$.
\begin{center}
\begin{tikzcd}
M(Y)\otimes N(Y)\arrow{r}{\theta_Y}\arrow{d}{r_f\otimes r_f} & P(Y)\arrow{d}{r_f}\\
M(X)\otimes N(X)\arrow{r}{\theta_X} & P(X)
\end{tikzcd}\\
\begin{tikzcd}
& M(X)\otimes N(X)\arrow{r}{\theta_X} & P(X)\arrow{dd}{t_f}\\
M(X)\otimes N(Y)\arrow{ru}{id\otimes r_f}\arrow{rd}{t_f\otimes id} & &\\
& M(Y)\otimes N(Y)\arrow{r}{\theta_Y} & P(Y)
\end{tikzcd}\\
\begin{tikzcd}
& M(X)\otimes N(X)\arrow{r}{\theta_X} & P(X)\arrow{dd}{t_f}\\
M(Y)\otimes N(X)\arrow{ru}{r_f\otimes id}\arrow{rd}{id\otimes t_f} & &\\
& M(Y)\otimes N(Y)\arrow{r}{\theta_Y} & P(Y)
\end{tikzcd}
\end{center}
\end{lem}
A good exposition and proof of this lemma can be found in \cite{Shulman}.
The data in this lemma is called a Dress paring.
The natural transformations from $M\square N$ to another Mackey functor $P$ are the same as the Dress parings from $M$ and $N$ to $P$, via the natural bijection $$\hom_{\A}(M\square N, P)\cong\text{Dress}(M, N; P),$$ that maps a map $\theta:M\square N\to P$ to $\theta_X:M(X)\otimes N(X)\to P(X\times X)\xrightarrow[]{r\Delta} P(X)$.
The first map comes from the Kan adjunction and $r\Delta$ is the restriction associated to the diagonal map of $G$-sets $X\to X\times X$.

In \A, the Burnside ring Mackey functor $$B^G(-) := \hom_{\Omega_R(G)}(-, G/G)$$ is the unit for box product, as shown in \cite{Lewis}.
In this way, $(\A, B^G(-), \square)$ is a symmetric monoidal category.

\section{The Box Product in $\B$}
Given two left $\mu_R(G)$-modules $M$ an $N$, we can form an $R$-module $A_H$ for each $H\leq G$ by induction on the cardinality of subgroups of $G$
\begin{equation*}\begin{split}
A_e & := (I^e_eM)\otimes_R(I^e_e N)\\
A_H & := \left((I^H_HM)\otimes_R(I^H_HN)\right) \oplus \bigoplus_{K<H}A_K.
\end{split}\end{equation*}
We say $A_H$ is of grade $H$.
By combining all the grades together, we get $A:=\oplus_{H\leq G}A_H$.
Note that the component $A_K$ in grade $H$ is distinct from the grade $K$.
Since $\mu_R(G)$ is generated by $R,I,C$'s, we can endow $A$ with a left $\mu_R(G)$-module structure by giving actions of $R,I,C$'s on $A$.
The maps $C_{g, H}, I_H^L, R^H_J$ map the grade $H$ to grades $^gH, L, J$ respectively, and map the other grades to zero.
Their action on the grade $H$ is described as follows:
\begin{enumerate}[(1)]
\item $I_H^L$ action on the grade $H$:\\
If $H = L$, $I_H^L$ acts as the identity on grade $H$.\\
If $H < L$, $I_H^L$ maps an element in $A_H$ to its corresponding copy $A_H$ in grade $L$.
To distinguish $A_H$ from its copy in grade $L$, we write its copy in grade $L$ as $I_H^L A_H$ from now on.
That is, grade $H$ is written as $$A_H = \left((I^H_H M)\otimes_R(I^H_H N)\right) \oplus \bigoplus_{K<H}I^H_K A_K.$$
\item $C_{g, H}$ action on the grade $H$:\\
We define the action of $C_{g, H}$ by induction on the cardinality of $H$ as follows.
For $m\otimes n\in (I^H_HM)\otimes (I^H_HN)$ and $x\in A_K$, where $K<H$,
\begin{equation*}\begin{split}
C_{g,H}(m\otimes n) &:= (C_{g,H}m)\otimes(C_{g,H}n)\in A_{^gH}\\
C_{g,H}I^H_K(x) &:= I^{^gH}_{^gK}C_{g, K}(x)\in A_{^gH}.
\end{split}\end{equation*}
\item $R^H_J$ action on the grade $H$:\\
We also define the action of $R_J^H$ by induction on the cardinality of $H$ as follows.
For $m\otimes n\in (I^H_HM)\otimes (I^H_HN)$ and $x\in A_K$, where $K<H$,
\begin{equation*}\begin{split}
R^H_J(m\otimes n) &:= (R^H_Jm)\otimes (R^H_Jn)\in A_J\\
R^H_JI^H_K(x) &:= \sum_{g\in [J\backslash H/K]} I_{^gK\cap J}^J C_g R^K_{K\cap J^g}(x)\in A_J.
\end{split}\end{equation*}
\end{enumerate}
\begin{defn}[Box product in \B]
\label{def:box_product_2}
Based on the left $\mu_R(G)$-module $\oplus_{H\leq G}A_H$, we can define $M\square N$ as $$M\square N:= \left(\oplus_{H\leq G}A_H\right)/FR,$$ where $FR$ is a submodule, called the Frobenius reciprocity submodule, generated by elements of the form
$$a\otimes (I_K^Hb) - I_K^H((R^H_Ka)\otimes b)$$
and $$(I_K^Hc)\otimes d - I_K^H(c\otimes (R_K^H d))$$ for all $K<H$, $a\in I^H_HM$, $b\in I^K_KN$, $c\in I^K_KM$ and $d\in I^H_HN$.
\end{defn}
Naturally, the image of $A_H$ under the quotient is called the grade $H$ of $M\square N$.

\begin{prop}
$\oplus_{H\leq G}\hom_{\Omega_R(G)}(G/H, G/G)$ is the unit for box product in \B.
\end{prop}
\begin{proof}
Let $M$ be a left $\mu_R(G)$-module and $N = \oplus_{H\leq G}\hom_{\Omega_R(G)}(G/H, G/G)$.
Then $I^H_HN$ is an $R$-module generated by $G/H\xleftarrow[]{\pi^H_L}G/L\to G/G = I^H_LR^H_Ln_H$, where $$n_H:=\left(G/H=G/H\to G/G\right).$$
Thus, $I^H_HM\otimes I^H_HN$ is a $R$-module generated by $m\otimes I^H_LR^H_Ln_H = I^H_L(R^H_Lm\otimes n_H)$.
Then we get a natural bijection between each grade of $M$ and $M\square N$
\begin{equation*}\begin{split}
F:I^H_HM&\longleftrightarrow I^H_H(M\square N)\\
m&\longmapsto m\otimes n_H\\
I^H_LR^H_Lm&\longmapsfrom I^H_L(R^H_Lm\otimes n_H)\in I^H_HM\otimes I^H_HN\\
I^H_KF^{-1}(x)&\longmapsfrom I^H_Kx\in I^H_KI^K_K(M\square N).
\end{split}\end{equation*}
Here $F^{-1}$ is defined by induction on the grade.
It is easy to check that this is a bijection and that it preserves the $\mu_R(G)$-module structure.
Thus, $M\square N$ is isomorphic to $M$ as a $\mu_R(G)$-module and $N$ is the unit for box product in \B.
\end{proof}
In this way, $(\B, \oplus_{H\leq G}\hom_{\Omega_R(G)}(G/H, G/G), \square)$ is a symmetric monoidal category.

\section{Equivalence of box product in $\A$ and \B}
\begin{thm}
\label{thm:main}
The equivalence of categories $\Phi:\A\xrightarrow[]{\cong}\B$ is a symmetric monoidal equivalence.
In other words, there are a natural isomorphism $\Phi M\square \Phi N \cong \Phi(M\square N)$ for any two Mackey functors $M, N\in\A$, and a compatible natural isomorphism $B^G(-)\xrightarrow[]{\cong}\hom_{\Omega_R(G)}(G/H, G/G)$.
\end{thm}
\begin{lem}
\label{lem:main_lem}
For any Mackey functors $M, N, P\in\A$, there is a natural bijection $$\text{Dress}(M, N; P)\cong\hom_{\B}(\Phi M\square\Phi N, \Phi P).$$
\end{lem}
\begin{proof}
Given $\beta\in\hom_{\B}(\Phi M\square\Phi N, \Phi P)$, we map it to $\theta\in\text{Dress}(M, N; P)$ defined as follows.
For each $H\leq G$, $\beta$ maps grade $I^H_H(\Phi M\square\Phi N)$ to grade $I^H_H\Phi P$, because $\beta(x) = \beta(I^H_Hx) = I^H_H\beta(x)\in I^H_H\Phi P$ for each $x\in I^H_H(\Phi M\square \Phi N)$.
Since $$I^H_H(\Phi M\square\Phi N) = (M(G/H)\otimes N(G/H))\oplus\bigoplus_{K<H}I_K^HA_K/FR$$ and $$I^H_H\Phi P = P(G/H),$$ $\beta$ induces an $R$-module homomorphism $\theta_{G/H}$ $$\theta_{G/H}:M(G/H)\otimes N(G/H)\to P(G/H)$$ by restricting to the first summand.
For a general finite $G$-set $X = \sqcup_{i=1}^p G/H_i$, $\theta_X$ is defined as 
\begin{equation*}\begin{split}
\theta_X:\bigoplus_{i,j=1}^p M(G/H_i)\otimes N(G/H_j)&\longrightarrow\bigoplus_{i=1}^pP(G/H_i)\\
m_i\otimes n_j & \longmapsto\left\{\begin{split}
\theta_{G/H_i}(m_i\otimes n_i) & \text{, if } i = j\\
0 & \text{, otherwise.}
\end{split}\right.
\end{split}\end{equation*}

Having constructed $\theta$, we now proceed to show that $\theta\in\text{Dress}(M, N; P)$.
Given finite $G$-sets $X,Y$ and a $G$-equivariant map $f:X\to Y$, it is sufficient to show that the three diagrams in Lemma \ref{lem:dress} commute.

If both $X$ and $Y$ are orbits, say $X = G/K$ and $Y = G/H$, observe that a $G$-equivariant map $f$ from $G/K$ to $G/H$ must be of the the form $f = \pi^H_{^gK}c_{g,K}$ for some $g\in G$.
By composition of commuting diagrams, we only need to consider the case where $f = c$ and $f = \pi$.

When $f = c_{g, H}:G/H\to G/^gH$, we have that $r_f = C_{g^{-1}, ^gH}$ and $t_f = C_{g, H}$.
The first diagram commutes because 
\begin{equation*}\begin{split}
r_f\theta_{Y}(m\otimes n)
& = C_{g^{-1}, ^gH}\beta(m\otimes n) = \beta(C_{g^{-1}, ^gH}(m\otimes n))\\
& = \beta(C_{g^{-1}, ^gH}m\otimes C_{g^{-1}, ^gH}n)) = \theta_{X}(r_fm\otimes r_fn).
\end{split}\end{equation*}
The second diagram commutes because
\begin{equation*}\begin{split}
t_f\theta_{X}(m\otimes r_fn)
& = C_{g,H}\beta(m\otimes C_{g^{-1}, ^gH}n) = \beta(C_{g,H}(m\otimes C_{g^{-1}, ^gH}n))\\
& = \beta(C_{g,H}m\otimes n) = \theta_{Y}(t_fm\otimes n).
\end{split}\end{equation*}
The third diagram commutes similarly.

When $f = \pi^H_K:G/K\to G/H$, we have that $r_f = R^H_K$ and $t_f = I_K^H$.
The first diagram commutes because 
\begin{equation*}\begin{split}
r_f\theta_{Y}(m\otimes n)
& = R^H_K\beta(m\otimes n) = \beta(R^H_K(m\otimes n))\\
& = \beta(R^H_Km\otimes R^H_Kn)) = \theta_{X}(r_fm\otimes r_fn).
\end{split}\end{equation*}
The second diagram commutes because
\begin{equation*}\begin{split}
t_f\theta_{X}(m\otimes r_fn)
& = I_K^H\beta(m\otimes R^H_Kn) = \beta(I_K^H(m\otimes R^H_Kn))\\
& = \beta(I_K^Hm\otimes n) = \theta_{Y}(t_fm\otimes n).
\end{split}\end{equation*}
The third diagram commutes similarly.

Now, if $Y$ is an orbit $G/H$ and $X = \sqcup_{i=1}^p G/K_i$ is a general $G$-set, denote the restriction of $f$ to $G/K_i$ by $f_i:G/K_i\to G/H$.
Thus, $r_f:M(G/H)\to\oplus_{i=1}^p M(G/K_i)$ is the sum of $r_{f_i}:M(G/H)\to M(G/K_i)$.
Similarly, $t_f:\oplus_{i=1}^p M(G/K_i)\to M(G/H)$ is determined by components $t_{f_i}:M(G/K_i)\to M(G/H)$.
The first diagram commutes because
\begin{equation*}\begin{split}
r_f\theta_{Y}(m\otimes n)
& = \sum_{i=1}^pr_{f_i}\theta_{Y}(m\otimes n) = \sum_{i=1}^p\theta_{X}(r_{f_i}m\otimes r_{f_i}n)\\
& = \theta_{X}(\sum_{i=1}^pr_{f_i}m\otimes r_{f_i}n) = \theta_{X}(\sum_{i=1}^pr_{f_i}m\otimes \sum_{i=1}^pr_{f_i}n)\\
& = \theta_{X}(r_fm\otimes r_fn).
\end{split}\end{equation*}
The second diagram commutes because
\begin{equation*}\begin{split}
t_f\theta_{\mathbf{b}}(m\otimes r_fn)
& = t_f\theta_{X}(\sum_{i=1}^p m_i\otimes \sum_{i=1}^pr_{f_i}n)
= t_f\theta_{X}(\sum_{i=1}^p m_i\otimes r_{f_i}n)\\
& = \sum_{i=1}^p t_f\theta_{X}(m_i\otimes r_{f_i}n)
= \sum_{i=1}^p t_{f_i}\theta_{G/K_i}(m_i\otimes r_{f_i}n)\\
& = \sum_{i=1}^p \theta_{Y}(t_{f_i}m_i\otimes n)
= \theta_{Y}(\sum_{i=1}^p t_{f_i}m_i\otimes n)\\
& = \theta_{Y}(t_f m\otimes n)
\end{split}\end{equation*}
for $m = \sum_{i=1}^p m_i$, where $m_i\in M(G/K_i)$.
The third diagram commutes similarly.

Lastly, the general case: $X = \sqcup_{i=1}^pX_i, Y = \sqcup_{i=1}^p G/H_i$, and $f$ maps $X_i$ to $G/H_i$.
Thus, $r_f$ maps $M(G/H_i)$ to $M(X_i)$ and $t_f$ maps $M(X_i)$ to $G/H_i$.
For the first diagram, take $m\in G/H_i$ and $n\in G/H_j$.
Thus, $r_f m\in M(X_i)$ and $r_f n\in M(X_j)$.
If $i\neq j$, then $\theta_{Y}(m\otimes n) = 0$ and $\theta_{X}(r_fm\otimes r_fn) = 0$.
If $i = j$, then this is the case when $Y$ is an orbit.
For the second diagram, take $m\in M(X_i)$ and $n\in G/H_j$.
Thus, $t_f m\in M(G/H_i)$ and $r_f n\in M(X_j)$.
If $i\neq j$, then $\theta_{X}(m\otimes r_fn) = 0$ and $\theta_{Y}(t_f m\otimes n) = 0$.
If $i = j$, it reduces to the case when $Y$ is an orbit.
The third diagram commutes similarly.

Given a Dress pairing $\theta\in\text{Dress}(M, N; P)$, we define $\beta$, a map from $\Phi M\square \Phi N$ to $P$ grade by grade, as follows:
\begin{equation*}\begin{split}
\beta_H:I^H_H(\Phi M\square\Phi N)&\longrightarrow I^H_H\Phi P = P(G/H)\\
M(G/H)\otimes N(G/H)\ni m\otimes n&\longmapsto \theta_{G/H}(m\otimes n)\\
I_K^HA_K\ni I^H_K(x)&\longmapsto I^H_K(\beta_K(x))
\end{split}\end{equation*}

Having constructed $\beta$, we now proceed to show that $\beta$ is indeed a map of $\mu_R(G)$-modules.
$\beta$ is linear in multiplication by elements in $\mu_R(G)$:
It is enough to check this for the generators $R, I, C$'s.
Say $K<H\leq G$ and take $m\otimes n\in M(G/H)\otimes N(G/H)$.
Then
\begin{equation*}\begin{split}
R^H_K\beta(m\otimes n) 
& = R^H_K\theta_{G/H}(m\otimes n) = \theta_{G/K}(R^H_K m\otimes R^H_K n)\\
& = \beta(R^H_K(m\otimes n))\\
C_{g, H}\beta(m\otimes n)
& = C_{g, H}\theta_{G/H}(m\otimes n) = \theta_{G/^gH}(C_{g, H}m\otimes C_{g, H}n)\\
& = \beta(C_{g, H}(m\otimes n)).
\end{split}\end{equation*}
Take $x\in A_K$ and $S\in\mu_R(G)$.
Since $SI^H_K$ acts on the grade $K$, $SI^H_K\beta(x) = \beta(SI^H_Kx)$ by induction.
Therefore, $$S\beta(I^H_Kx) = SI^H_K(x) = \beta(SI^H_Kx).$$

Let us show that $\beta$ maps the Frobenius reciprocity submodule $FR_H$ to zero for each $H\leq G$.
Take $K<H$, $a\in M(G/H)$ and $b\in N(G/K)$.
By the second commuting diagram in lemma \ref{lem:dress}, we have 
\begin{equation*}\begin{split}
\beta(I^H_K(R^H_Ka\otimes b)) 
& = I^H_K\beta(R^H_Ka\otimes b) = I^H_K\theta_{G/K}(R^H_Ka\otimes b)\\ 
& = \theta_{G/H}(a\otimes I^H_K b) = \beta(a\otimes I^H_K b).
\end{split}\end{equation*}
Take $K<H$, $c\in M(G/K)$ and $d\in N(G/H)$.
By the third commuting diagram in lemma \ref{lem:dress}, we have
\begin{equation*}\begin{split}
\beta(I^H_K(c\otimes R^H_Kd)) 
& = I^H_K\beta(c\otimes R^H_Kd)  = I^H_K\theta_{G/K}(c\otimes R^H_Kd)\\
& = \theta_{G/H}(I^H_K c\otimes d) = \beta(I^H_Kc\otimes d).
\end{split}\end{equation*}

Lastly, it is easy to see the composition of those two maps above is identity in either way.
For instance, the map $$\text{Dress}(M, N; P)\to \hom_{\B}(\Phi M\square\Phi N, \Phi P)\to \text{Dress}(M, N; P)$$, which maps $\theta\mapsto\beta\mapsto\theta'$, is identity because $\theta'_{G/H}$ equals to the restriction of $\beta$ to the first summand of the grade $H$, which in turn equals to the maps $\theta_{G/H}$ according to the constructions above.
Thus, $\theta' = \theta$.
\end{proof}

\begin{proof}[Proof for Theorem \ref{thm:main}]
Fix three Mackey functors $M,N,P\in\A$.
By Lemma \ref{lem:main_lem}, there is a natural bijection $$\text{Dress}(M, N; P)\cong\hom_{\B}(\Phi M\square\Phi N, \Phi P).$$
By Lemma \ref{lem:dress}, there is a natural bijection $$\hom_{\A}(M\square N, P)\cong\text{Dress}(M, N; P).$$
Since $\Phi:\A\xrightarrow[]{\cong}\B$ is an equivalence of categories, there is a natural bijection $$\hom_{\A}(M\square N, P)\cong\hom_{\B}(\Phi (M\square N), \Phi P).$$
Therefore, we get a natural bijection $$\hom_{\B}(\Phi M\square\Phi N, \Phi P)\cong\hom_{\B}(\Phi (M\square N), \Phi P).$$
Therefore, $\Phi M\square \Phi N$ is naturally isomorphic to $\Phi (M\square N)$ as a left $\mu_R(G)$-module.

Moreover, the equality $\Phi(B^G(-)) = \oplus_{H\leq G}B^G(G/H) = \oplus_{H\leq G}\hom_{\Omega_R(G)}(G/H, G/G)$ verifies the correspondence of units for box products in $\A$ and $\B$.
Thus, the equivalence $\Phi$ is monoidal.
\end{proof}

\begin{thebibliography}{9}
\bibitem{Green}
J.A. Green,
\textit{Axiomatic representation theory for finite groups},
1971.

\bibitem{Dress}
Andreas W.M. Dress,
\textit{Contributions to the theory of induced representations},
1973.

\bibitem{Lindner}
Harald Lindner,
\textit{A remark on Mackey-functors},
1976.

\bibitem{Lewis}
L. Gaunce Lewis,
\textit{The theory of Green functors},
1981.

\bibitem{Thevenaz}
Jacques Th\'{e}venaz and Peter Webb,
\textit{The structure of Mackey functors},
1991.

\bibitem{Shulman}
Megan Elizabeth Shulman,
\textit{Equivariant local coefficients and the $RO(G)$-graded cohomology of classifying spaces},
2010.

\bibitem{Mazur}
Kristen Luise Mazur,
\textit{On the structure of Mackey functors and Tambara functors},
2013.
\end{thebibliography}

\end{document}
1509.07051.extracted_bib
{Green J.A. Green, Axiomatic representation theory for finite groups, 1971.
1509.07051.extracted_bib
{Dress Andreas W.M. Dress, Contributions to the theory of induced representations, 1973.
1509.07051.extracted_bib
{Lindner Harald Lindner, A remark on Mackey-functors, 1976.
1509.07051.extracted_bib
{Lewis L. Gaunce Lewis, The theory of Green functors, 1981.
1509.07051.extracted_bib
{Thevenaz Jacques Th\'{evenaz and Peter Webb, The structure of Mackey functors, 1991.
1509.07051.extracted_bib
{Shulman Megan Elizabeth Shulman, Equivariant local coefficients and the $RO(G)$-graded cohomology of classifying spaces, 2010.
1509.07051.extracted_bib
{Mazur Kristen Luise Mazur, On the structure of Mackey functors and Tambara functors, 2013.