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Recall that any extensive category can be canonically endowed the structure of a site via the extensive topology, which is the Grothendieck topology whose covering morphisms are the coproduct injections.

I've heard that ($Cat$-valued) stacks for the extensive topology on an extensive category $C$ are precisely the pseudofunctors which send coproducts in $C$ to products in $Cat$. Is there an explicit proof of this anywhere?

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This a genral fact: the coproducts in an extensive category are disjoint so the sheaf/stack condition with respect to a cover by the coproduct injection exactly say that the coproduct is sent to a product. So if they are the only cover a sheaf or a stack is just a functor sending the coproduct to products.

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