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Corollary 4.6.4.10. Let $q: X \rightarrow S$ be an isofibration of simplicial sets, let $g: B \rightarrow S$ be a morphism of simplicial sets, and let $f_0, f_1: B \rightarrow X$ be morphisms satisfying $q \circ f_0 = g = q \circ f_1$. Let $A$ be a simplicial subset of $B$ satisfying $f_0|_{A} = f_1|_{A}$, and let

\[ (B \coprod _{A} B) \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]

be a categorical mapping cylinder of $B$ relative to $A$. The following conditions are equivalent:

$(a)$

The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}_{A/ \, /S}(B, X)$.

$(b)$

The lifting problem

\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ (s_0, s_1)} \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B} \ar [r]^-{ g \circ \pi } \ar@ {-->}[ur]^{ \overline{f} } & S } \]

admits a solution.

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 Theorem 4.6.4.8, and the implication $(b) \Rightarrow (a)$ from the implication $(2) \Rightarrow (1)$ of Theorem 4.6.4.8. $\square$