Remark 7.1.4.32. In the situation of Corollary 7.1.4.31, suppose we are given a functor of $\infty $-categories $F: \operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{D}}$. The following conditions are equivalent:
- $(1)$
The functor $F$ preserves $K$-indexed colimits.
- $(2)$
The functor $(F \circ L): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $K$-indexed colimits, where $L: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}'$ denotes a left adjoint to the inclusion functor.
The implication $(1) \Rightarrow (2)$ is clear (the functor $L$ preserves $K$-indexed colimits by Corollary 7.1.4.28). Conversely, suppose that condition $(2)$ is satisfied and let $\overline{q}': K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}'$ be a colimit diagram; we wish to show that $F \circ \overline{q}'$ is a colimit diagram in $\operatorname{\mathcal{D}}$. Set $q = \overline{q}'|_{K}$. Since $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits, we can extend $q$ to a colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Then $\overline{q}'$ is isomorphic to the composition $L \circ \overline{q}$, so $F \circ \overline{q}'$ is isomorphic to $(F \circ L) \circ \overline{q}$ and is therefore a colimit diagram by virtue of assumption $(2)$.