# Kerodon

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Corollary 6.3.3.14. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories. Then $F$ is an equivalence if and only if it satisfies the following pair of conditions:

$(1)$

The functor $F$ is conservative. That is, a morphism $u$ of $\operatorname{\mathcal{C}}$ is an isomorphism if and only if $F(u)$ is an isomorphism in $\operatorname{\mathcal{D}}$.

$(2)$

The functor $F$ admits a fully faithful right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

Proof. Suppose that conditions $(1)$ and $(2)$ are satisfied; we will show that $F$ is an equivalence of $\infty$-categories (the converse is immediate from the definitions). Combining assumption $(2)$ with Proposition 6.3.3.13, we can choose a functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ and a natural isomorphism $\epsilon : F \circ G \xrightarrow {\sim } \operatorname{id}_{\operatorname{\mathcal{D}}}$ which is the counit of an adjunction between $F$ and $G$. Let $\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F$ be a natural transformation which is compatible up to homotopy with $\epsilon$, in the sense of Definition 6.2.1.1. We will complete the proof by showing that $\eta$ is also an isomorphism. Fix an object $C \in \operatorname{\mathcal{C}}$, we wish to show that the map $\eta _{C}: C \rightarrow (G \circ F)(C)$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{C}}$ (Theorem 4.4.4.4). By virtue of assumption $(1)$, it will suffice to show that $F( \eta _{C} )$ is an isomorphism in the $\infty$-category $\operatorname{\mathcal{D}}$. The compatibility of $\eta$ and $\epsilon$ guarantees that the diagram

$\xymatrix { & (F \circ G \circ F)(C) \ar [dr]^{ \epsilon _{F(C)} } & \\ F(C) \ar [ur]^{ F( \eta _ C ) } \ar [rr]^{\operatorname{id}} & & F(C) }$

commutes in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{D}}}$. Since $\epsilon _{F(C)}$ is an isomorphism, it follows that $F( \eta _ C )$ is also an isomorphism (Remark 1.3.6.3). $\square$