Image (category theory)

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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: #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''. ''See also'': *universal property *subobject *coimage *image (mathematics)