Proposition 7.7.2.26. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let $F: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ be a morphism. If $F$ is a colimit diagram, then it is a weak colimit diagram. The converse holds if $\operatorname{\mathcal{C}}$ admits finite products or if $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits.
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Proof. The first assertion follows from the observation that the forgetful functor $U: \operatorname{\mathcal{C}}_{/C} \rightarrow \operatorname{\mathcal{C}}$ is conservative and creates colimits (Proposition 7.1.4.20). To prove the converse, it will suffice to show that the functor $U$ preserves $K$-indexed colimits. If $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits, this follows from Corollary 7.1.4.21. If $\operatorname{\mathcal{C}}$ admits finite products, then the functor $U$ has a right adjoint (given on objects by the construction $X \mapsto X \times C$; see Proposition 7.6.1.14) and therefore preserves $K$-indexed colits by virtue of Corollary 7.1.4.22. $\square$