  Let $k$ be an algebraically closed field of positive characteristic $p$, and
$\mathbb{F}$ be an algebraically closed field of characteristic 0. We consider
the $\mathbb{F}$-linear category $\mathbb{F} pp_k^\Delta$ of finite groups, in
which the set of morphisms from $G$ to $H$ is the $\mathbb{F}$-linear extension
$\mathbb{F} T^\Delta(H,G)$ of the Grothendieck group $T^\Delta(H,G)$ of
$p$-permutation $(kH,kG)$-bimodules with (twisted) diagonal vertices. The
$\mathbb{F}$-linear functors from $\mathbb{F} pp_k^\Delta$ to
$\mathbb{F}\hbox{-Mod}$ are called {\em diagonal $p$-permutation functors}.
They form an abelian category $\mathcal{F}_{pp_k}^\Delta$.
  We study in particular the functor $\mathbb{F}T^{\Delta}$ sending a finite
group $G$ to the Grothendieck group $\mathbb{F}T(G)$ of $p$-permutation
$kG$-modules, and show that $\mathbb{F}T^\Delta$ is a semisimple object of
$\mathcal{F}_{pp_k}^\Delta$, equal to the direct sum of specific simple
functors parametrized by isomorphism classes of pairs $(P,s)$ of a finite
$p$-group $P$ and a generator $s$ of a $p'$-subgroup acting faithfully on $P$.
This leads to a precise description of the evaluations of these simple
functors. In particular, we show that the simple functor indexed by the trivial
pair $(1,1)$ is isomorphic to the functor sending a finite group $G$ to
$\mathbb{F} K_0(kG)$, where $K_0(kG)$ is the group of projective $kG$-modules.
