Remark 7.1.3.13. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $K$ be a simplicial set, and let $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ be a limit diagram with restriction $u = \overline{u}|_{K}$. The following conditions are equivalent:
- $(1)$
The composition $(F \circ \overline{u}): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram.
- $(2)$
For every limit diagram $\overline{u}': K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ with $\overline{u}'|_{K} = u$, the composition $(F \circ \overline{u}'): K^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram.
The implication $(2) \Rightarrow (1)$ is immediate. For the converse, we observe that if $\overline{u}': K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is another limit diagram with $\overline{u}'|_{K} = u$, then $\overline{u}$ and $\overline{u}'$ are isomorphic when viewed as objects of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/u}$, so that $F \circ \overline{u}$ and $F \circ \overline{u}'$ are isomorphic when viewed as objects of the $\infty $-category $\operatorname{\mathcal{D}}_{/ (F \circ u)}$. Since $F \circ \overline{u}$ is a final object of $\operatorname{\mathcal{D}}_{/ (F \circ u)}$, it follows that $F \circ \overline{u}'$ is also a final object of $\operatorname{\mathcal{D}}_{ / (F \circ u)}$ (Corollary 4.6.7.15).