  Let $k$ be an algebraically closed field of characteristic $p>0$, let $R$ be
a commutative ring and let $\mathcal{F}$ be an algebraically closed field of
characteristic $0$. We introduce the category $\overline{\mathcal{F}_{Rpp_k}}$
of stable diagonal $p$-permutation functors over $R$. We prove that the
category $\overline{\mathcal{F}_{\mathbb{F}pp_k}}$ is semisimple and give a
parametrization of its simple objects in terms of the simple diagonal
$p$-permutation functors.
  We also introduce the notion of a stable functorial equivalence over $R$
between blocks of finite groups. We prove that if $G$ is a finite group and if
$b$ is a block idempotent of $kG$ with an abelian defect group $D$ and
Frobenius inertial quotient $E$, then there exists a stable functorial
equivalence over $\mathbb{F}$ between the pairs $(G,b)$ and $(D\rtimes E,1)$.
