# Kerodon

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Remark 6.2.1.10. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty$-categories which admits a right adjoint $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$. The existence of natural transformations

$\eta : \operatorname{id}_{\operatorname{\mathcal{C}}} \rightarrow G \circ F \quad \quad \epsilon : F \circ G \rightarrow \operatorname{id}_{\operatorname{\mathcal{D}}}$

guarantee that $F$ and $G$ are simplicial homotopy inverses of one another, in the sense of Definition 3.1.6.1. In particular, $F$ and $G$ are homotopy equivalences of simplicial sets.