  Let $V_*\otimes V\rightarrow\mathbb{C}$ be a non-degenerate pairing of
countable-dimensional complex vector spaces $V$ and $V_*$. The Mackey Lie
algebra $\mathfrak{g}=\mathfrak{gl}^M(V,V_*)$ corresponding to this paring
consists of all endomorphisms $\varphi$ of $V$ for which the space $V_*$ is
stable under the dual endomorphism $\varphi^*: V^*\rightarrow V^*$. We study
the tensor Grothendieck category $\mathbb{T}$ generated by the
$\mathfrak{g}$-modules $V$, $V_*$ and their algebraic duals $V^*$ and $V^*_*$.
This is an analogue of categories considered in prior literature, the main
difference being that the trivial module $\mathbb{C}$ is no longer injective in
$\mathbb{T}$. We describe the injective hull $I$ of $\mathbb{C}$ in
$\mathbb{T}$, and show that the category $\mathbb{T}$ is Koszul. In addition,
we prove that $I$ is endowed with a natural structure of commutative algebra.
We then define another category $_I\mathbb{T}$ of objects in $\mathbb{T}$ which
are free as $I$-modules. Our main result is that the category ${}_I\mathbb{T}$
is also Koszul, and moreover that ${}_I\mathbb{T}$ is universal among abelian
$\mathbb{C}$-linear tensor categories generated by two objects $X$, $Y$ with
fixed subobjects $X'\hookrightarrow X$, $Y'\hookrightarrow Y$ and a pairing
$X\otimes Y\rightarrow \text{\textbf{1}}$ where \textbf{1} is the monoidal
unit. We conclude the paper by discussing the orthogonal and symplectic
analogues of the categories $\mathbb{T}$ and ${}_I\mathbb{T}$.
