Remark 7.1.7.8 (05QJ)

Remark 7.1.7.8. Let $K$ be a simplicial set, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $K$-indexed colimits, and let $\operatorname{Fun}'( K^{\triangleright }, \operatorname{\mathcal{C}})$ be the full subcategory of $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}})$ spanned by the colimit diagrams. It follows from Corollary 7.1.7.7 that the restriction map

\[ \operatorname{Fun}'( K^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \quad \quad \overline{f} \mapsto \overline{f}|_{K} \]

admits a section $s: \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}'( K^{\triangleright }, \operatorname{\mathcal{C}})$. Let $v$ denote the cone point of $K^{\triangleright }$. Then the composition

\[ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \xrightarrow {s} \operatorname{Fun}'( K^{\triangleright }, \operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_{v} } \operatorname{\mathcal{C}} \]

is a functor $T$ which carries each diagram $f: K \rightarrow \operatorname{\mathcal{C}}$ to a colimit $T(f)$ in the $\infty $-category $\operatorname{\mathcal{C}}$. In fact, we can be more precise: $T$ is a colimit functor for $\operatorname{\mathcal{C}}$, in the sense of Definition 7.1.1.17. More precisely, we observe that precomposition with the tautological map

\[ q: \Delta ^1 \times K \simeq K \star _{K} K \rightarrow K \star _{ \Delta ^0 } \Delta ^0 \simeq K^{\triangleright } \]

determines a map

\[ \eta : \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \xrightarrow {s} \operatorname{Fun}'( K^{\triangleright }, \operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}}) \xrightarrow { \circ q} \operatorname{Fun}(\Delta ^1 \times K, \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}( \Delta ^1, \operatorname{Fun}(K,\operatorname{\mathcal{C}}) ), \]

carrying each diagram $f$ to a natural transformation $\eta _{f}: f \rightarrow \underline{ T(f) }$ which exhibits $T(f)$ as a colimit of $f$ (Remark 7.1.3.6). It follows from the criterion of Corollary 6.2.4.5 that $\eta $ exhibits $T$ as a left adjoint of the diagonal map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$.