Kerodon

Proposition 7.1.7.2. Let $K$ be a simplicial set and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $K$-indexed colimits. Then, for every simplicial set $B$, the $\infty $-category $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$ also admits $K$-indexed colimits. Moreover, a morphism of simplicial sets $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ is a colimit diagram if and only if it is a levelwise colimit diagram (Definition 7.1.7.1). In particular, for every vertex $b \in B$, the evaluation functor $\operatorname{ev}_{b}: \operatorname{Fun}(K,\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ preserves $K$-indexed colimits.