Corollary 4.6.3.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$ be a pair of diagrams indexed by a simplicial set $B$. Let
\[ (B \coprod B) \xrightarrow {(s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]
be a categorical mapping cylinder for $B$ (Definition 4.6.3.3). The following conditions are equivalent:
- $(a)$
The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$.
- $(b)$
There exists a diagram $\overline{f}: \overline{B} \rightarrow \operatorname{\mathcal{C}}$ satisfying $f_0 = \overline{f} \circ s_0$ and $f_1 = \overline{f} \circ s_1$.
In particular, condition $(b)$ does not depend on the choice of categorical mapping cylinder.