Lemma 9.4.6.11. Let $K$ be a simplicial set and let $\widehat{U}: \widehat{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}$ be a cartesian fibration of simplicial sets having the property that, for each vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\widehat{\operatorname{\mathcal{E}}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \widehat{\operatorname{\mathcal{E}}}$ admits $K$-indexed colimits. Let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits $K$-indexed colimits, let $F: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ be a diagram, and let $S$ be the collection of vertices $C \in \operatorname{\mathcal{C}}$ for which the functor $F_{C} = F|_{ \operatorname{\mathcal{E}}_{C} }$ preserves $K$-indexed colimits. Then $S$ is closed under retracts in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.
Proof. Using Corollary 5.6.7.3, we can reduce to the case where $\operatorname{\mathcal{C}}$ is an $\infty $-category. Fix objects $C, C' \in \operatorname{\mathcal{C}}$ such that $C$ is a retract of $C'$, and assume that $F_{C'}$ preserves $K$-indexed colimits; we wish to show that $F_{C}$ also preserves $K$-indexed colimits. Choose a colimit diagram $\overline{u}: K^{\triangleright } \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C}$; we will show that $F_{C} \circ \overline{u}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. Set $u = \overline{u}|_{K}$. It follows from Theorem 5.2.1.1 that $\widehat{U}$ induces a cartesian fibration $\operatorname{Fun}( K, \widehat{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Applying Remark 8.5.1.23, we deduce that there is a diagram $u': K \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C'}$ having the property that $u$ is a retract of $u'$ in the $\infty $-category $\operatorname{Fun}(K, \widehat{\operatorname{\mathcal{E}}} )$. Since $\widehat{\operatorname{\mathcal{E}}}_{C'}$ admits $K$-indexed colimits, we can extend $u'$ to a colimit diagram $\overline{u}': K^{\triangleright } \rightarrow \widehat{\operatorname{\mathcal{E}}}_{C'}$. Since $\widehat{U}$ is a cartesian fibration, $\overline{u}$ and $\overline{u}'$ are $\widehat{U}$-colimit diagrams in the $\infty $-category $\widehat{\operatorname{\mathcal{E}}}$ (Corollary 7.3.3.23). Using Theorem 7.3.6.14, we see that any diagram which exhibits $u$ as a retract of $u'$ can be extended to a diagram which exhibits $\overline{u}$ as a retract of $\overline{u}'$. It follows that $F_{C} \circ \overline{u}$ is a retract of $F_{C'} \circ \overline{u}'$ (in the $\infty $-category $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{D}})$). Consequently, to show that $F_{C} \circ \overline{u}$ is a colimit diagram in $\operatorname{\mathcal{D}}$, it will suffice to show that $F_{C'} \circ \overline{u}'$ is a colimit diagram in $\operatorname{\mathcal{D}}$ (Corollary 8.5.1.12). This follows from our assumption that $F_{C'}$ preserves $K$-indexed colimits. $\square$