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In SGA1 Expose 13 it says: (I'm translating) Let $S$ be a scheme, and $f: X \rightarrow Y$ a morphism of $S$-schemes. Definition 1.1: Let $F$ be a stack over $X$. We will say that $(F,f)$ is cohomologically proper relative to $S$ in dimension $\leq -1$ (resp. in dimension $\leq 0$, resp. in dimension $\leq 1$) if, for every $S$-scheme $S'$ ($g:S'\rightarrow S$), the canonical functor: $g^*f_*F \rightarrow f'_*h^*F$ (where $h$ is the canonical $h: X\times_S S' \rightarrow X$ and $f'$ is the base change of $f$ to $S'$) is faithful (resp. fully faithful, resp. an equivalence of categories).

This seems to suggest that they had a more general idea in mind of what "cohomologically proper" means. What is it?

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It is studied by Illusie in his appendix to Deligne's Théorèmes de finitude en cohomologie $\ell$-adique, SGA 4 1/2. The notion is introduced in 1.3 of this appendix.

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