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Remark 4.6.4.5. Let $i: A \hookrightarrow B$ be a monomorphism of simplicial sets, and let $(\operatorname{id}_ B, \operatorname{id}_ B): (B \coprod _{A} B) \rightarrow B$ be the fold map. Unwinding the definitions, we see that a categorical mapping cylinder for $B$ relative to $A$ can be identified with a factorization of $(\operatorname{id}_ B, \operatorname{id}_ B)$ as a composition

\[ B \coprod _{A} B \xrightarrow {\iota } \overline{B} \xrightarrow {\pi } B, \]

where $\iota $ is a monomorphism of simplicial sets and $\pi $ is a categorical equivalence. Such factorizations always exist: by virtue of Exercise 3.1.7.11, we can even arrange that $\pi $ is a trivial Kan fibration of simplicial sets (hence a categorical equivalence by virtue of Proposition 4.5.2.9).