Essentially surjective functor

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Definition

Symbol-free definition

A functor is termed essentially surjective if every object in the target category is isomorphic to the image under the functor of some object in the source category.

Definition with symbols

A functor ${\mathcal {F}}:{\mathcal {C}}\to {\mathcal {D}}$ is termed essentially surjective' if for any $B\in {\mathcal {D}}$, there exists $A\in {\mathcal {C}}$ such that ${\mathcal {F}}(A)$ is isomorphic to $B$.