The Stacks project

Lemma 7.28.4. Let $\mathcal{C}$, $\mathcal{D}$ be sites. Let $u : \mathcal{C} \to \mathcal{D}$ be a cocontinuous functor. Let $U$ be an object of $\mathcal{C}$, and set $V = u(U)$. We have a commutative diagram

\[ \xymatrix{ \mathcal{C}/U \ar[r]_{j_ U} \ar[d]_{u'} & \mathcal{C} \ar[d]^ u \\ \mathcal{D}/V \ar[r]^-{j_ V} & \mathcal{D} } \]

where the left vertical arrow is $u' : \mathcal{C}/U \to \mathcal{D}/V$, $U'/U \mapsto V'/V$. Then $u'$ is cocontinuous also and we get a commutative diagram of topoi

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

where $f$ (resp. $f'$) corresponds to $u$ (resp. $u'$).

Proof. The commutativity of the first diagram is clear. It implies the commutativity of the second diagram provided we show that $u'$ is cocontinuous.

Let $U'/U$ be an object of $\mathcal{C}/U$. Let $\{ V_ j/V \to u(U')/V\} _{j \in J}$ be a covering of $u(U')/V$ in $\mathcal{D}/V$. Since $u$ is cocontinuous there exists a covering $\{ U_ i' \to U'\} _{i \in I}$ such that the family $\{ u(U_ i') \to u(U')\} $ refines the covering $\{ V_ j \to u(U')\} $ in $\mathcal{D}$. In other words, there exists a map of index sets $\alpha : I \to J$ and morphisms $\phi _ i : u(U_ i') \to V_{\alpha (i)}$ over $U'$. Think of $U_ i'$ as an object over $U$ via the composition $U'_ i \to U' \to U$. Then $\{ U'_ i/U \to U'/U\} $ is a covering of $\mathcal{C}/U$ such that $\{ u(U_ i')/V \to u(U')/V\} $ refines $\{ V_ j/V \to u(U')/V\} $ (use the same $\alpha $ and the same maps $\phi _ i$). Hence $u' : \mathcal{C}/U \to \mathcal{D}/V$ is cocontinuous. $\square$


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