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

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

commutes.

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 (7.25.8.1) 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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