Corollary 4.6.3.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams indexed by a simplicial set $B$, and let $A \subseteq B$ be a simplicial subset satisfying $f_0|_{A} = f_1|_{A}$. The following conditions are equivalent:
- $(1)$
The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}_{A/}(B, \operatorname{\mathcal{C}})$.
- $(2)$
There exists a factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ as a composition
\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B, \]where $\pi $ is a categorical equivalence, and a morphism $\overline{f}: \overline{B} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.
- $(3)$
For every factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ as a composition
\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]where the map $(s_0, s_1): B \coprod _{A} B \rightarrow \overline{B}$ is a monomorphism, there exists a morphism $\overline{f}: \overline{B} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.