Lemma 7.21.7. Let $\mathcal{C}$ and $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a functor. Assume that

1. $u$ is cocontinuous,

2. $u$ is continuous, and

3. $u$ is fully faithful.

For $g_!, g^{-1}, g_*$ as above the canonical maps $\mathcal{F} \to g^{-1}g_!\mathcal{F}$ and $g^{-1}g_*\mathcal{F} \to \mathcal{F}$ are isomorphisms for all sheaves $\mathcal{F}$ on $\mathcal{C}$.

Proof. Let $X$ be an object of $\mathcal{C}$. In Lemmas 7.20.2 and 7.21.5 we have seen that sheafification is not necessary for the functors $g^{-1} = (u^ p\ )^\#$ and $g_{*} = ({}_ pu\ )^\#$. We may compute $(g^{-1}g_{*}\mathcal{F})(X) = g_{*}\mathcal{F}(u(X)) = \mathop{\mathrm{lim}}\nolimits \mathcal{F}(Y)$. Here the limit is over the category of pairs $(Y, u(Y) \to u(X))$ where the morphisms $u(Y) \to u(X)$ are not required to be of the form $u(\alpha )$ with $\alpha$ a morphism of $\mathcal{C}$. By assumption (c) we see that they automatically come from morphisms of $\mathcal{C}$ and we deduce that the limit is the value on $(X, u(\text{id}_ X))$, i.e., $\mathcal{F}(X)$. This proves that $g^{-1}g_{*}\mathcal{F} = \mathcal{F}$.

On the other hand, $(g^{-1}g_{!}\mathcal{F})(X) = g_{!}\mathcal{F}(u(X)) = (u_ p\mathcal{F})^\# (u(X))$, and $u_ p\mathcal{F}(u(X)) = \mathop{\mathrm{colim}}\nolimits \mathcal{F}(Y)$. Here the colimit is over the category of pairs $(Y, u(X) \to u(Y))$ where the morphisms $u(X) \to u(Y)$ are not required to be of the form $u(\alpha )$ with $\alpha$ a morphism of $\mathcal{C}$. By assumption (c) we see that they automatically come from morphisms of $\mathcal{C}$ and we deduce that the colimit is the value on $(X, u(\text{id}_ X))$, i.e., $\mathcal{F}(X)$. Thus for every $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ we have $u_ p\mathcal{F}(u(X)) = \mathcal{F}(X)$. Since $u$ is cocontinuous and continuous any covering of $u(X)$ in $\mathcal{D}$ can be refined by a covering (!) $\{ u(X_ i) \to u(X)\}$ of $\mathcal{D}$ where $\{ X_ i \to X\}$ is a covering in $\mathcal{C}$. This implies that $(u_ p\mathcal{F})^+(u(X)) = \mathcal{F}(X)$ also, since in the colimit defining the value of $(u_ p\mathcal{F})^+$ on $u(X)$ we may restrict to the cofinal system of coverings $\{ u(X_ i) \to u(X)\}$ as above. Hence we see that $(u_ p\mathcal{F})^+(u(X)) = \mathcal{F}(X)$ for all objects $X$ of $\mathcal{C}$ as well. Repeating this argument one more time gives the equality $(u_ p\mathcal{F})^\# (u(X)) = \mathcal{F}(X)$ for all objects $X$ of $\mathcal{C}$. This produces the desired equality $g^{-1}g_!\mathcal{F} = \mathcal{F}$. $\square$

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