Corollary 8.5.1.11. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be a functor of $\infty $-categories, let $K$ be a simplicial set, and suppose we are given a pair of diagrams $f,g: K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$. If $f$ is a $U$-colimit diagram and $g$ is a retract of $f$ (in the $\infty $-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}})$), then $g$ is also a $U$-colimit diagram.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Corollary 4.1.3.3, we can choose an inner anodyne morphism $K \hookrightarrow \operatorname{\mathcal{K}}$, where $\operatorname{\mathcal{K}}$ is an $\infty $-category. Using Remark 4.3.6.7, we see that the induced map $K^{\triangleright } \hookrightarrow \operatorname{\mathcal{K}}^{\triangleright }$ is also inner anodyne. We may therefore extend $f$ and $g$ to functors $F,G: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$. Since the restriction functor $\operatorname{Fun}( \operatorname{\mathcal{K}}^{\triangleright }, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{D}})$ is a trivial Kan fibration (Proposition 1.5.7.6), it follows that $G$ is a retract of $F$. By virtue of Corollary 7.2.2.2, we can replace $K$ by $\operatorname{\mathcal{K}}$ and thereby reduce to proving Corollary 8.5.1.11 in the special case where $K$ is an $\infty $-category. In this case, the desired result is a special case of Corollary 8.5.1.10 (see Example 7.3.3.9). $\square$