Kerodon

Remark 4.5.1.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories. If $\operatorname{\mathcal{D}}$ is a Kan complex, then $\operatorname{\mathcal{C}}$ is a Kan complex. To prove this, it suffices to show that every morphism $u: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$ is an isomorphism (Proposition 4.4.2.1). By virtue of Remark 4.5.1.20, this is equivalent to the assertion that $F(u): F(X) \rightarrow F(Y)$ is an isomorphism in $\operatorname{\mathcal{D}}$, which is automatic when $\operatorname{\mathcal{D}}$ is a Kan complex (Proposition 1.4.6.10). Similarly, if $\operatorname{\mathcal{C}}$ is a Kan complex, then $\operatorname{\mathcal{D}}$ is a Kan complex (this follows by applying the same argument to an inverse equivalence $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$).