The Stacks project

Proof. Given an abelian sheaf $\mathcal{G}$ on $\mathcal{C}'$ the underlying sheaf of sets of $f^{-1}\mathcal{G}$ is the same as $f^{-1}$ of the underlying sheaf of sets of $\mathcal{G}$, see Sites, Section 7.44. Hence the exactness of $f^{-1}$ for sheaves of sets (required in the definition of a morphism of topoi, see Sites, Definition 7.15.1) implies the exactness of $f^{-1}$ as a functor on abelian sheaves.

To see the statement on modules recall that $f^*\mathcal{F}$ is defined as the tensor product $f^{-1}\mathcal{F} \otimes _{f^{-1}\mathcal{O}', f^\sharp } \mathcal{O}$. Hence $f^*$ is a composition of functors both of which are exact. $\square$

Proof. Say $f$ is given by the continuous functor $u : \mathcal{D} \to \mathcal{C}$. We have to show that the restriction of the map to $\mathcal{C}/U$ for any $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is an isomorphism. We may replace $U$ by the members of a covering of $U$. Hence by Sites, Lemma 7.14.10 we may assume there exists a morphism $U \to u(V)$ for some $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Of course, then we may replace $U$ by $u(V)$. Then since $u$ is continuous, we may replace $V$ by a covering and assume there is a presentation $\mathcal{O}_ V^{\oplus m} \to \mathcal{O}_ V^{\oplus n} \to \mathcal{F}|_ V \to 0$ over $\mathcal{D}/V$. Since formation of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ commutes with localization (Lemma 18.27.2) we may replace $f$ by the morphism $(\mathcal{C}/u(V), \mathcal{O}_{u(V)}) \to (\mathcal{D}/V, \mathcal{O}_ V)$ induced by $f$. Hence we reduce to the case where $\mathcal{F}$ has a global presentation $\mathcal{O}_\mathcal {D}^{\oplus m} \to \mathcal{O}_\mathcal {D}^{\oplus n} \to \mathcal{F} \to 0$. Since $f$ is flat and $f^*\mathcal{O}_\mathcal {D} = \mathcal{O}_\mathcal {C}$ we obtain a corresponding presentation $\mathcal{O}_\mathcal {C}^{\oplus m} \to \mathcal{O}_\mathcal {C}^{\oplus n} \to f^*\mathcal{F} \to 0$, see Lemma 18.31.2. Using that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ commutes with finite direct sums in the first variable, using that both $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {C}}(\mathcal{O}_\mathcal {C}, -)$ and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{O}_\mathcal {D}, -)$ are the identity functor, and using the functoriality of the construction of Remark 18.27.3 we obtain a commutative diagram

where the right two vertical arrows are isomorphisms. By Lemma 18.27.5 the rows are exact. We conclude by the 5 lemma. $\square$