The Stacks project

Proof. By the Yoneda lemma (Categories, Lemma 4.3.5) the sheaf $\mathcal{G}_ V$ corresponding to $V \in E$ is defined up to unique isomorphism by the formula $f^{-1}\mathcal{F}(V) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}_ V, \mathcal{F})$. Recall that $f^{-1}\mathcal{F}(V) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(h_ V^\# , f^{-1}\mathcal{F})$. Denote $i_ V : h_ V^\# \to f^{-1}\mathcal{G}_ V$ the map corresponding to $\text{id}$ in $\mathop{\mathrm{Mor}}\nolimits (\mathcal{G}_ V, \mathcal{G}_ V)$. Functoriality in (1) implies that the bijection is given by

For any $V_1, V_2 \in E$ there is a canonical map

which is characterized by $f^{-1}(f_!(\varphi )) \circ i_{V_2} = i_{V_1} \circ \varphi$. Note that $\varphi \mapsto f_!(\varphi )$ is compatible with composition; this can be seen directly from the characterization. Hence $h_ V^\# \mapsto \mathcal{G}_ V$ and $\varphi \mapsto f_!\varphi$ is a functor from the full subcategory of $\mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ whose objects are the $h_ V^\#$.

Let $J$ be a set and let $J \to E$, $j \mapsto V_ j$ be a map. Then we have a functorial bijection

using the product of the bijections above. Hence we can extend the functor $f_!$ to the full subcategory of $\mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ whose objects are coproducts of $h_ V^\#$ with $V \in E$.

Given an arbitrary sheaf $\mathcal{H}$ on $\mathcal{D}$ we choose a coequalizer diagram

where $\mathcal{H}_ i = \coprod h_{V_{i, j}}^\#$ is a coproduct with $V_{i, j} \in E$. This is possible by assumption (2), see Lemma 7.12.5 (for those worried about set theoretical issues, note that the construction given in Lemma 7.12.5 is canonical). Define $f_!(\mathcal{H})$ to be the sheaf on $\mathcal{C}$ which makes

a coequalizer diagram. Then

Hence we see that we can extend $f_!$ to the whole category of sheaves on $\mathcal{D}$. $\square$