Kerodon

Remark 7.1.4.18. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a conservative functor of $\infty $-categories and let $u: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Suppose that $F$ creates $K$-indexed limits and that $F \circ u$ can be extended to a limit diagram $K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$. Then an extension $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ of $u$ is a limit diagram if and only if $F \circ \overline{u}$ is a limit diagram in $\operatorname{\mathcal{D}}$ (see Lemma 7.1.4.16).