Kerodon

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Corollary 3.5.1.28 (Transitivity). Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be morphisms of simplicial sets and let $n$ be an integer. Then:

$(1)$: If $f$ and $g$ are $n$-connective, then the composition $(g \circ f): X \rightarrow Z$ is $n$-connective.
$(2)$: If $g \circ f$ is $n$-connective and $g$ is $(n+1)$-connective, then $f$ is $n$-connective.
$(3)$: If $f$ is $n$-connective and $(g \circ f)$ is $(n+1)$-connective, then $g$ is $(n+1)$-connective.

Proof. Using Proposition 3.1.7.1, we can reduce to the case where $Z$ is a Kan complex and the morphisms $f$ and $g$ are Kan fibrations. Using the criterion of Proposition 3.5.1.22, we can further reduce to the case $Z = \Delta ^0$. In this case, Corollary 3.5.1.28 is a restatement of Proposition 3.5.1.26. $\square$