The Stacks project

Lemma 21.38.4. Assumptions and notation as in Situation 21.38.3. For $U' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ set $U = u(U')$ and $V = p'(U')$ and consider the induced morphisms of ringed topoi

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'/U'), \mathcal{O}_{U'}) \ar[rd]_{\pi '_{U'}} \ar[rr]_{g'} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_ U) \ar[ld]^{\pi _ U} \\ & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V), \mathcal{O}_ V) } \]

Then there exists a morphism of topoi

\[ \sigma ' : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'/U'), \]

such that setting $\sigma = g' \circ \sigma '$ we have $\pi '_{U'} \circ \sigma ' = \text{id}$, $\pi _ U \circ \sigma = \text{id}$, $(\sigma ')^{-1} = \pi '_{U', *}$, and $\sigma ^{-1} = \pi _{U, *}$.

Proof. Let $v' : \mathcal{D}/V \to \mathcal{C}'/U'$ be the functor constructed in the proof of Lemma 21.38.2 starting with $p' : \mathcal{C}' \to \mathcal{D}'$ and the object $U'$. Since $u$ is a $1$-morphism of fibred categories over $\mathcal{D}$ it transforms strongly cartesian morphisms into strongly cartesian morphisms, hence the functor $v = u \circ v'$ is the functor of the proof of Lemma 21.38.2 relative to $p : \mathcal{C} \to \mathcal{D}$ and $U$. Thus our lemma follows from that lemma. $\square$

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