Definition:Image (Category Theory)

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Let $\CC$ be a locally small category.

Let $f : X \to Y$ be a morphism in $\CC$.

An image of $f$ consists of an object $I$ and a monomorphism $m: I \to Y$ such that:

$(1): \quad$ There exists a morphism $e : X \to I$ such that $f = m \circ e$.
$(2): \quad$ For any object $I'$ with a morphism $e' : X \to I'$ and a monomorphism $m' : I' \to Y$ such that $f = m' \circ e'$, there exists a unique morphism $v : I \to I'$ such that $m = m' \circ v$.