Kerodon

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Definition 4.6.2.9. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. The essential image of $F$ is the full subcategory of $\operatorname{\mathcal{D}}$ spanned by those objects $D \in \operatorname{\mathcal{D}}$ for which there exists an object $C \in \operatorname{\mathcal{C}}$ and an isomorphism $F(C) \simeq D$. We say that $F$ is essentially surjective if its essential image is the entire $\infty $-category $\operatorname{\mathcal{D}}$: that is, if the map of sets $\pi _0( \operatorname{\mathcal{C}}^{\simeq } ) \rightarrow \pi _0( \operatorname{\mathcal{D}}^{\simeq } )$ is surjective.