# The Stacks Project

## Tag 03EH

Lemma 7.24.7. 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 cocontinuous functors. Here $j : \mathcal{C}/V \to \mathcal{C}/U$, $(a : W \to V) \mapsto (f \circ a : W \to U)$ 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 7.20.2 we see that the following diagram of morphisms of topoi $$\tag{7.24.7.1} \vcenter{ \xymatrix{ \mathop{\textit{Sh}}\nolimits(\mathcal{C}/V) \ar[rd]_{j_V} \ar[rr]_j & & \mathop{\textit{Sh}}\nolimits(\mathcal{C}/U) \ar[ld]^{j_U} \\ & \mathop{\textit{Sh}}\nolimits(\mathcal{C}) & } }$$ 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. $\square$

The code snippet corresponding to this tag is a part of the file sites.tex and is located in lines 4845–4864 (see updates for more information).

\begin{lemma}
\label{lemma-relocalize}
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 cocontinuous functors. Here $j : \mathcal{C}/V \to \mathcal{C}/U$,
$(a : W \to V) \mapsto (f \circ a : W \to U)$
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 $j_{V*} = j_{U*} \circ j_*$.
\end{lemma}

\begin{proof}
The commutativity of the diagram is immediate.
The agreement of $j$ with $j_{V/U}$ follows from the definitions. By
Lemma \ref{lemma-composition-cocontinuous}
we see that the following diagram of morphisms of topoi

\label{equation-relocalize}
\vcenter{
\xymatrix{
\Sh(\mathcal{C}/V) \ar[rd]_{j_V} \ar[rr]_j & &
\Sh(\mathcal{C}/U) \ar[ld]^{j_U} \\
& \Sh(\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 $j_{V!} = j_{U!} \circ j_!$
\end{proof}

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