Lemma 7.25.8. Let $\mathcal{C}$ be a site. Let $f : V \to U$ be a morphism of $\mathcal{C}$. Then there exists a commutative diagram

$\xymatrix{ \mathcal{C}/V \ar[rd]_{j_ V} \ar[rr]_ j & & \mathcal{C}/U \ar[ld]^{j_ U} \\ & \mathcal{C} & }$

of continuous and cocontinuous functors. The functor $j : \mathcal{C}/V \to \mathcal{C}/U$, $(a : W \to V) \mapsto (f \circ a : W \to U)$ is identified with the functor $j_{V/U} : (\mathcal{C}/U)/(V/U) \to \mathcal{C}/U$ via the identification $(\mathcal{C}/U)/(V/U) = \mathcal{C}/V$. Moreover we have $j_{V!} = j_{U!} \circ j_!$, $j_ V^{-1} = j^{-1} \circ j_ U^{-1}$, and $j_{V*} = j_{U*} \circ j_*$.

Proof. The commutativity of the diagram is immediate. The agreement of $j$ with $j_{V/U}$ follows from the definitions. By Lemma 7.21.2 we see that the following diagram of morphisms of topoi

7.25.8.1
$$\label{sites-equation-relocalize} \vcenter { \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V) \ar[rd]_{j_ V} \ar[rr]_ j & & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U) \ar[ld]^{j_ U} \\ & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) & } }$$

is commutative. This proves that $j_ V^{-1} = j^{-1} \circ j_ U^{-1}$ and $j_{V*} = j_{U*} \circ j_*$. The equality $j_{V!} = j_{U!} \circ j_!$ follows formally from adjointness properties. $\square$

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