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Suppose $f:X\longrightarrow Y$ is a morphism of schemes. Take the categories $\mathbf{X}_{et},\,\mathbf{Y}_{et}$ of étale morphisms over $X$ and $Y$. Then is the direct image functor:

$f_{*}:\mathbf{PSh}(\mathbf{X}_{et})\longrightarrow\mathbf{PSh}(\mathbf{Y}_{et})$

on category of presheaves exact? I know it's left-exact if restricted to the category of sheaves on étale site, but what does exactness mean in the category of presheaves on a site?

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    $\begingroup$ Exactness means what it usually does, i.e. preservation of short exact sequences. A short exact sequence of presheaves is just a short exact sequence at every value. $\endgroup$ – Kevin Carlson Mar 18 '17 at 17:09
  • $\begingroup$ Yes but in case of sheaves because the etale site is small, we have a good characterisation of epimorphism: ncatlab.org/nlab/show/category+of+sheaves#EpiMonoIsomorphisms $\endgroup$ – Pavle Papunashvili Mar 18 '17 at 20:19
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    $\begingroup$ Ah, but an epimorphism of presheaves is just a map which is an epimorphism at every level-exactly dual to a monomorphism. $\endgroup$ – Kevin Carlson Mar 19 '17 at 1:38
  • $\begingroup$ Ah, thanks. But is the functor exact? It came up in the proof that $Rf_{*}G(U)$ is the sheaf associated to $H_{et}^q(U,G))$ and without exactness of $f_{*}$ the proof falls apart. $\endgroup$ – Pavle Papunashvili Mar 19 '17 at 7:20
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The answer to your question is yes.

Exactness for presheaves on a site means that for every object $U$ in the topology, applying the functor $\Gamma(U,-|_U)$ is exact. The direct image of a presheaf $\mathscr F$ is the presheaf $U\mapsto\Gamma(U_X,-|_{U_X})$, where $U_X=U\times_Y X$. It's now tautological that direct image is an exact functor on presheaves.

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