Remark 4.8.7.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then:
If $F$ is categorically $1$-connective and $\operatorname{\mathcal{C}}$ is a Kan complex, then $\operatorname{\mathcal{D}}$ is also a Kan complex.
If $F$ is categorically $2$-connective, then $\operatorname{\mathcal{C}}$ is a Kan complex if and only if $\operatorname{\mathcal{D}}$ is a Kan complex.