Proposition 4.5.1.22. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The following conditions are equivalent:
- $(1)$
The functor $F$ is an equivalence of $\infty $-categories.
- $(2)$
For every simplicial set $X$, composition with $F$ induces an equivalence of $\infty $-categories $\operatorname{Fun}(X, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{D}})$.
- $(3)$
For every simplicial set $X$, composition with $F$ induces a homotopy equivalence of Kan complexes $\operatorname{Fun}(X, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{D}})^{\simeq }$.
- $(4)$
For every $\infty $-category $\operatorname{\mathcal{B}}$, composition with $F$ induces a homotopy equivalence of Kan complexes $\operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{C}})^{\simeq } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})^{\simeq }$.
- $(5)$
For every $\infty $-category $\operatorname{\mathcal{B}}$, composition with $F$ induces a bijection of sets $\pi _0( \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})^{\simeq } )$.