The Stacks project

Lemma 7.28.3. Let $f : \mathcal{C} \to \mathcal{D}$ be a morphism of sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. Let $V \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{D})$, $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and $c : U \to u(V)$ a morphism of $\mathcal{C}$. There exists a commutative diagram of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[r]_{j_ U} \ar[d]_{f_ c} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \ar[d]^ f \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) \ar[r]^{j_ V} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}). } \]

We have $f_ c = f' \circ j_{U/u(V)}$ where $f' : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/u(V)) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V)$ is as in Lemma 7.28.1 and $j_{U/u(V)} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/u(V))$ is as in Lemma 7.25.8. Using the identifications $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) = \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_ U^\# $ and $\mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V) = \mathop{\mathit{Sh}}\nolimits (\mathcal{D})/h_ V^\# $ of Lemma 7.25.4 the functor $(f_ c)^{-1}$ is described by the rule

\[ (f_ c)^{-1}(\mathcal{H} \xrightarrow {\varphi } h_ V^\# ) = (f^{-1}\mathcal{H} \times _{f^{-1}\varphi , h_{u(V)}^\# , c} h_ U^\# \rightarrow h_ U^\# ). \]

Finally, given any morphisms $b : V' \to V$, $a : U' \to U$ and $c' : U' \to u(V')$ such that

\[ \xymatrix{ U' \ar[r]_-{c'} \ar[d]_ a & u(V') \ar[d]^{u(b)} \\ U \ar[r]^-c & u(V) } \]

commutes, then the diagram

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U') \ar[r]_{j_{U'/U}} \ar[d]_{f_{c'}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[d]^{f_ c} \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V') \ar[r]^{j_{V'/V}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}/V). } \]


Proof. This lemma proves itself, and is more a collection of things we know at this stage of the development of theory. For example the commutativity of the first square follows from the commutativity of Diagram ( and the commutativity of the diagram in Lemma 7.28.1. The description of $f_ c^{-1}$ follows on combining Lemma 7.25.9 with Lemma 7.28.1. The commutativity of the last square then follows from the equality

\[ f^{-1}\mathcal{H} \times _{h_{u(V)}^\# , c} h_ U^\# \times _{h_ U^\# } h_{U'}^\# = f^{-1}(\mathcal{H} \times _{h_ V^\# } h_{V'}^\# ) \times _{h_{u(V'), c'}^\# } h_{U'}^\# \]

which is formal using that $f^{-1}h_ V^\# = h_{u(V)}^\# $ and $f^{-1}h_{V'}^\# = h_{u(V')}^\# $, see Lemma 7.13.5. $\square$

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