# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

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 $\pi _0( \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{C}})^{\simeq } ) \rightarrow \pi _0( \operatorname{Fun}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{D}})^{\simeq } )$.

Proof. The implication $(1) \Rightarrow (2)$ follows from Remark 4.5.1.16, the implication $(2) \Rightarrow (3)$ from Remark 4.5.1.19, the implication $(3) \Rightarrow (4)$ is immediate, and the implication $(4) \Rightarrow (5)$ follows from Remark 3.1.6.5, and the implication $(5) \Rightarrow (1)$ follows from Yoneda's lemma (applied to the homotopy category $\mathrm{h} \mathit{\operatorname{QCat}}$). $\square$