Remark 4.8.9.6 (Transitivity). Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be morphisms of simplicial sets and let $n$ be an integer.
- $(1)$
Suppose that $f$ and $g$ are categorically $n$-connective. Then $g \circ f$ is categorically $n$-connective.
- $(2)$
Suppose that $g \circ f$ is categorically $n$-connective, $g$ is categorically $(n+1)$-connective, and $n \geq 1$. Then $f$ is categorically $n$-connective.
- $(3)$
Suppose that $g \circ f$ is categorically $n$-connective and that $f$ is categorically $(n-1)$-connective. Then $g$ is categorically $n$-connective.
To prove these assertions, we can use Remark 4.8.9.4 to reduce to the case where $A$, $B$, and $C$ are $\infty $-categories, in which case the result follows from Proposition 4.8.7.12