Lemma 7.21.2. Let $u : \mathcal{C} \to \mathcal{D}$, and $v : \mathcal{D} \to \mathcal{E}$ be cocontinuous functors. Then $v \circ u$ is cocontinuous and we have $h = g \circ f$ where $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$, resp. $g : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{E})$, resp. $h : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{E})$ is the morphism of topoi associated to $u$, resp. $v$, resp. $v \circ u$.

**Proof.**
Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $\{ E_ i \to v(u(U))\} $ be a covering of $U$ in $\mathcal{E}$. By assumption there exists a covering $\{ D_ j \to u(U)\} $ in $\mathcal{D}$ such that $\{ v(D_ j) \to v(u(U))\} $ refines $\{ E_ i \to v(u(U))\} $. Also by assumption there exists a covering $\{ C_ l \to U\} $ in $\mathcal{C}$ such that $\{ u(C_ l) \to u(U)\} $ refines $\{ D_ j \to u(U)\} $. Then it is true that $\{ v(u(C_ l)) \to v(u(U))\} $ refines the covering $\{ E_ i \to v(u(U))\} $. This proves that $v \circ u$ is cocontinuous. To prove the last assertion it suffices to show that ${}_ sv \circ {}_ su = {}_ s(v \circ u)$. It suffices to prove that ${}_ pv \circ {}_ pu = {}_ p(v \circ u)$, see Lemma 7.20.2. Since ${}_ pu$, resp. ${}_ pv$, resp. ${}_ p(v \circ u)$ is right adjoint to $u^ p$, resp. $v^ p$, resp. $(v \circ u)^ p$ it suffices to prove that $u^ p \circ v^ p = (v \circ u)^ p$. And this is direct from the definitions.
$\square$

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