Kerodon

Remark 10.2.6.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $X_{\bullet }$ be an augmented simplicial object of $\operatorname{\mathcal{C}}$, so that $F(X_{\bullet } )$ is an augmented cosimplicial object of $\operatorname{\mathcal{D}}$. Composition with the functor $F$ carries splittings of $X_{\bullet }$ to splittings of $F( X_{\bullet } )$. Consequently, if $X_{\bullet }$ is split, then $F( X_{\bullet } )$ is also split. In particular, if $X_{\bullet }$ is split, then $F( X_{\bullet } )$ is a colimit diagram in $\operatorname{\mathcal{D}}$ (Proposition 10.2.6.9).