Exercise 3.1.7.11. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. Show that $f$ can be factored as a composition $X_{} \xrightarrow {f'} P_{}(f) \xrightarrow {f''} Y_{}$, where $f'$ is a monomorphism and $f''$ is a trivial Kan fibration. Moreover, this factorization can be chosen to depend functorially on $f$ (as an object of the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$).
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