Remark 4.8.6.7 (Transitivity). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories and let $n$ be an integer. Then:
- $(a)$
If $F$ and $G$ are essentially $n$-categorical, then the composite functor $G \circ F$ is essentially $n$-categorical.
- $(b)$
If $G \circ F$ is essentially $n$-categorical and $G$ is essentially $(n+1)$-categorical, then $F$ is essentially $n$-categorical.
- $(c)$
If $G \circ F$ is essentially $n$-categorical and $F$ is essentially $(n-1)$-categorical, full, and essentially surjective, then $G$ is essentially $n$-categorical.
Assertion $(a)$ follows from Remark 4.8.5.15, assertion $(b)$ from Proposition 4.8.5.31 (together with Exercise 4.8.5.32 in the case $n \leq -2$), and assertion $(c)$ follows from Proposition 4.8.5.33 (together with Remark 4.5.1.18 in the case $n \leq -2$).