# Kerodon

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Proposition 7.1.6.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $B$ be a simplicial set, and let $f: K \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be a diagram. Assume that, for every vertex $b \in B$, the composite diagram

$K \xrightarrow {f} \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}}$

admits a colimit in the $\infty$-category $\operatorname{\mathcal{C}}$. Then:

$(1)$

The diagram $f$ admits a colimit in $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$.

$(2)$

Let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be an extension of $f$. Then $\overline{f}$ is a colimit diagram if and only if, for every vertex $b \in B$, the morphism

$K^{\triangleright } \xrightarrow { \overline{f} } \operatorname{Fun}(B, \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}}$

is a colimit diagram in $\operatorname{\mathcal{C}}$.

Proof of Proposition 7.1.6.1. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category, let $B$ be a simplicial set, and let $f: K \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be a diagram. Assume that, for every vertex $b \in B$, the composite diagram

$K \xrightarrow {f} \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}}$

admits a colimit in the $\infty$-category $\operatorname{\mathcal{C}}$. Applying Corollary 7.1.6.8, we see that $f$ admits an extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ with the property that, for every vertex $b \in B$, the composition $\operatorname{ev}_{b} \circ \overline{f}$ is a colimit diagram in $\operatorname{\mathcal{C}}$. Applying Corollary 7.1.6.12, we see any such extension is a colimit diagram in $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$. To complete the proof, it will suffice to show the converse: if $\overline{f}': K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ is any colimit diagram extending $f$ and $b \in B$ is a vertex, then $\operatorname{ev}_{b} \circ \overline{f}'$ is also a colimit diagram in $\operatorname{\mathcal{C}}$. In this case, the extension $\overline{f}'$ is isomorphic to $\overline{f}$ as an object of the $\infty$-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{Fun}(B,\operatorname{\mathcal{C}}) )$. It follows that $\operatorname{ev}_{b} \circ \overline{f}'$ is isomorphic to $\operatorname{ev}_{b} \circ \overline{f}$ as an object of the $\infty$-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ and therefore a colimit diagram by virtue of Corollary 7.1.2.14. $\square$