# Image (category theory)

Given a category C and a morphism [itex]f:X\rightarrow Y[itex] in C, the image of f is a monomorphism [itex]h:I\rightarrow Y[itex] satisfying the following:

1. There exists a morphism [itex]g:X\rightarrow I[itex] such that f = hg.
2. For any object Z with a morphism [itex]k:X\rightarrow Z[itex] and a monomorphism [itex]l:Z\rightarrow Y[itex] such that f = lk, there exists a unique morphism [itex]m:I\rightarrow Z[itex] such that k = mg and h = lm.

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