# Morphism decomposition

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.