Corollary 7.1.3.15 (02K2)

Corollary 7.1.3.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and suppose we are given a pair of morphisms $u,v: K \rightarrow \operatorname{\mathcal{C}}$ which are isomorphic as objects of the $\infty $-category $\operatorname{Fun}(K,\operatorname{\mathcal{C}})$. Then:

$(1)$: The morphism $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ if and only if $v$ can be extended to a limit diagram $\overline{v}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$.
$(2)$: The morphism $u$ can be extended to a colimit diagram $\overline{u}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ if and only if $v$ can be extended to a colimit diagram $\overline{v}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

Proof. We will prove $(1)$; the proof of $(2)$ is similar. Suppose that $u$ can be extended to a limit diagram $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$. Since the diagrams $u$ and $v$ are isomorphic, it follows from Corollary 4.4.5.3 that $\overline{u}$ is isomorphic to a diagram $\overline{v}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ satisfying $\overline{v}|_{K} = v$. Applying Corollary 7.1.3.14, we conclude that $\overline{v}$ is also a limit diagram. $\square$