Let $G$ be an algebraic group over a field $k$ (say of characteristic $0$) and let $H,H'$ be two closed subgroups. I would like to understand the category $Rep_k(H \cap H')$ of finite dimensional representations of $H \cap H'$.

Here was my thought : by the tannakian formalism I think that, since $H \cap H' = H \times_G H'$ that $Rep_k(H\cap H')$ should be $Rep_k(H)\coprod_{Rep_k(G)} Rep_k(H')$ the pushout of $Rep_k(H)$ and $Rep_k(H')$ over $Rep_k(G)$ in the category of neutral tannakian categories with a fixed fiber functor.

And my first guess to describe this category would be that the objects would be elements of either $Rep_k(H)$ or $Rep_k(H')$ modulo the equivalence relation $V \sim V'$ if there exists $V_0 \in Rep_k(G)$ such that $V_{0|H} = V$ and $V_{0|H'} = V'$.

But that seems weird and probably false. Does anyone know the answer ?