Given a category C and a morphism <math>f:X\rightarrow Y</math> in C, the image of f is a monomorphism <math>h:I\rightarrow Y</math> satisfying the following universal property: There exists a morphism <math>g:X\rightarrow I</math> such that f = hg.

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  • Given a category C and a morphism <math>f:X\rightarrow Y</math> in C, the image of f is a monomorphism <math>h:I\rightarrow Y</math> satisfying the following universal property: There exists a morphism <math>g:X\rightarrow I</math> such that f = hg. For any object Z with a morphism <math>k:X\rightarrow Z</math> and a monomorphism <math>l:Z\rightarrow Y</math> such that f = lk, there exists a unique morphism <math>m:I\rightarrow Z</math> such that k = mg and h = lm. The image of f is often denoted by im f or Im(f). One can show that a morphism f is monic if and only if f = im f.
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  • Given a category C and a morphism <math>f:X\rightarrow Y</math> in C, the image of f is a monomorphism <math>h:I\rightarrow Y</math> satisfying the following universal property: There exists a morphism <math>g:X\rightarrow I</math> such that f = hg.
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  • Image (category theory)
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