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Corollary 7.2.1.14. Let $f: X \rightarrow Z$ be a morphism of simplicial sets. Then $f$ is left cofinal if and only if it factors as a composition $X \xrightarrow {f'} Y \xrightarrow {f''} Z$, where $f'$ is left anodyne and $f''$ is a trivial Kan fibration.

Proof. Suppose first that we can write $f = f'' \circ f'$, where $f'$ is left anodyne and $f''$ is a trivial Kan fibration. Proposition 7.2.1.3 guarantees that $f'$ is left cofinal, and Proposition 7.2.1.5 guarantees that $f''$ is left cofinal. Applying Proposition 7.2.1.6, we conclude that $f$ is also left cofinal.

We now prove the converse. Assume that $f: X \rightarrow Z$ is left cofinal. Applying Proposition 4.2.4.8, we can write $f$ as a composition $X \xrightarrow {f'} Y \xrightarrow {f''} Z$, where $f'$ is left anodyne and $f''$ is a left fibration. Then $f'$ is also left cofinal (Proposition 7.2.1.3). Applying Proposition 7.2.1.6, we deduce that $f''$ is left cofinal. It then follows from Corollary 7.2.1.13 that $f''$ is a trivial Kan fibration. $\square$