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Corollary 4.6.4.9. 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 $f_1 = \overline{f} \circ s_1$.

Proof. Apply Theorem 4.6.4.8 in the special case where $S = \Delta ^{0}$. $\square$