Kerodon

Definition 7.1.3.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, and let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Suppose that $q$ can be extended to a limit diagram $\overline{q}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. We say that the limit of $q$ is preserved by $F$ if the composition $F \circ \overline{q}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. Similarly, if $q$ can be extended to a colimit diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$, we say that the colimit of $q$ is preserved by $F$ if $F \circ \overline{q}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$.