Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.6.4.7. 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 4.6.4.3). 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.

Proof. The implication $(a) \Rightarrow (b)$ follows from the implication $(1) \Rightarrow (3)$ of Proposition 4.6.4.1, and the implication $(b) \Rightarrow (a)$ from the implication $(2) \Rightarrow (1)$ of Proposition 4.6.4.1. $\square$