Kerodon

Remark 9.5.6.7 (Colimits in Bousfield Localizations). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a Bousfield localization functor between presentable $\infty $-categories and let $K$ be a small simplicial set. Then a morphism $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ is a colimit diagram if and only if it is isomorphic to $F \circ \overline{p}$, where $\overline{p}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$. The “if” direction is immediate (since the functor $F$ preserves small colimits). For the converse, it suffices to show that $q = \overline{q}|_{K}$ is isomorphic to $F \circ p$ for some diagram $K \rightarrow \operatorname{\mathcal{C}}$ (we can then take $\overline{p}$ to be a colimit diagram extending $p$). For example, we can take $p = G \circ q$, where $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is right adjoint to $F$.