An abelian category, $A$, is said to satisfy axiom AB6 if for every family of filtered diagrams $I_j$ indexed by a set $J$, the canonical map $\mathrm{colim}_{\prod_{I_j}}(\prod_j M_{ij}) \to \prod_J \mathrm{colim}_{I_j} M_{ij}$ is an isomorphism for every collection of functors $M_{ij}\colon I_j \to A$.

Let X be a site. Is it true that $\mathrm{Shv}(X,\mathrm{Ab})$ satisfies AB6? I tried to look for a reference, but I didn't find any, even though I know that $AB4^*$ is not satisfied in general. Moreover, it seems to me that it should be true for a locally compact space, but I am not sure even about that.

An abelian category, $A$, is said to satisfy axiom AB6 if for every family of filtered diagrams $I_j$ indexed by a set $J$, the canonical map $\mathrm{colim}_{\prod_{I_j}}(\prod_j M_{ij}) \to \prod_J \mathrm{colim}_{I_j} M_{ij}$ is an isomorphism for every collection of functors $M_{ij}\colon I_j \to A$.

Let X be a site. Is it true that $\mathrm{Shv}(X,\mathrm{Ab})$ satisfies AB6? I tried to look for a reference, but I didn't find any, even though I know that AB$4^*$ is not satisfied in general. Moreover, it seems to me that it should be true for a locally compact space, but I am not sure even about that.