Let $\mathcal{C}$ and $\mathcal{D}$ be categories. A functor $F\colon \mathcal{C}\to \mathcal{D}$ is essentially surjective if for any object $A\in\mathcal{OB}(\mathcal{D})$ there exists an object $X\in\mathcal{OB}(\mathcal{C})$ such that $F(X)\cong A$ That is, there are morphisms (in $D$ $f \colon F(X)\to A$ and $g\colon A\to F(X)$ such that $fg=1_A$ and $gf=1_{F(X)}$

Remarks.

• Clearly, if $F$ is surjective, it is essentially surjective. But the reverse is not true.
• A functor is an equivalence iff it is full, faithful and essentially surjective.
• isomorphism-dense subcategory. A full subcategory $\mathcal{S}$ of a category $\mathcal{C}$ is said to be isomorphism-dense in $\mathcal{C}$ if the inclusion functor $\mathcal{S}\hookrightarrow \mathcal{C}$ is essentially surjective. Since $\mathcal{S}$ is full, the inclusion functor is full and faithful. As a result, $\mathcal{S}$ is isomorphism-dense if the inclusion functor is an equivalence.