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The category Set has the property that any morphism f can be decomposed into an epimorphism $e$ and a monomorphism $m$ with $f$ = $m\circ e$. (The intermediary set object is the image of $f$.)

Is there a name for this construction?

Is there a name for this property of categories?

In Set, the intermediary object is unique up to isomorphism. Is this the case for all categories with the property?

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The name of this idea is factorization system and a lot is known about them. See the online available book "Abstract and Concrete Categories", chapter IV, for example. A trivial counterexample to the statement that the intermediate object is unique: the poset with elements $A,B$ where $A \leq B$, you can factor the morphism $f : A \to B$ as $f \circ \operatorname{id}$ and $\operatorname{id} \circ f$ yet $A$ and $B$ are not isomorphic.

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A related notion is that of an exact category. That's what led me to this mathoverflow post, which could be exactly what you are looking for. In your further search efforts, it might help to search for "every morphism factors as an epimorphism followed by a monomorphism" - it seems to be a more widespread formulation of the property you describe.

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