Inverse image functor for sheaves on the etale site

Question

Let $f: X\to Y$ be an etale map and $\mathcal G,\mathcal F$ be sheaves on the etale sites of $X,Y$ respectively. We know that there exists an adjoint pair $f^*,f_*$ between these sites. I would like to show that $f^*$ is simply restriction. That is, I would like to prove that: $$\operatorname{Hom}_X(\mathcal F|_X,\mathcal G) \cong \operatorname{Hom}_Y(\mathcal F,f_*G).$$ Giving a map on the left hand side is the same as giving maps $\mathcal F(U\to X) \to \mathcal G(U \to X)$ for each etale map $U\to X$.

On the other hand, giving a map on the right hand side is the same as giving maps $\mathcal F(V \to Y) \to \mathcal G(V\times_Y X \to X)$ for each etale map $V\to Y$. These do not look like the same thing.

In the Zariski topology, this works because if there is an open inclusion $V\to X$, then $V\times_Y X = V$ but I don't think this is correct for arbitrary etale maps...

Answer 1

Even if $V\times_Y X\neq V$ in general, there exist an open immersion (hence étale) map $V\rightarrow V\times_Y X$. This gives your isomorphism.

More precisely, let $\alpha:\mathcal{F}_{|X}\rightarrow\mathcal{G}$. As you said, for every étale map $g:U\rightarrow X$, you have an morphism $$\alpha_g:\mathcal{F}_{|X}(U\rightarrow X)=\mathcal{F}(U\rightarrow X\rightarrow Y)\rightarrow\mathcal{G}(U\rightarrow X)$$ You now want to construct a map $\mathcal{F}\rightarrow f_*\mathcal{G}$. This will be the following : $$\mathcal{F}(V\rightarrow Y)\rightarrow\mathcal{F}(V\times_Y X\rightarrow X\rightarrow Y)=\mathcal{F}_{|X}(V\times_Y X\rightarrow X)\overset{\alpha}\rightarrow\mathcal{G}(V\times_Y X\rightarrow X)$$ where the first map is the restriction.

Conversely, let $\beta:\mathcal{F}\rightarrow f_*\mathcal{G}$. As you said,$\beta$ is the data of mophisms $$\mathcal{F}(V\rightarrow Y)\rightarrow\mathcal{G}(V\times_Y X\rightarrow X)$$ You want to construct a map $\mathcal{F}_{|X}\rightarrow\mathcal{G}$. This will be the following : $$\mathcal{F}_{|X}(U\rightarrow X)=\mathcal{F}(U\rightarrow X\rightarrow Y)\overset{\beta}\rightarrow\mathcal{G}(U\times_Y X\rightarrow X)\rightarrow\mathcal{G}(U\rightarrow X)$$ where the last map is given by the restriction along $U\rightarrow U\times_Y X$ which is étale.

I let you check that these constructions are indeed inverse to each other.