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essentially surjective
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.

isomorphismdense subcategory. A full subcategory $\mathcal{S}$ of a category $\mathcal{C}$ is said to be isomorphismdense 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 isomorphismdense if the inclusion functor is an equivalence.
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