The Stacks project

Remark 7.5.5. Suppose that $\mathcal{A}$ is a category such that any diagram $\mathcal{I}_ Y \to \mathcal{A}$ has a colimit in $\mathcal{A}$. In this case it is clear that there are functors $u^ p$ and $u_ p$, defined in exactly the same way as above, on the categories of presheaves with values in $\mathcal{A}$. Moreover, the adjointness of the pair $u^ p$ and $u_ p$ continues to hold in this setting.

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