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:

1. There exists a morphism <math>g:X\rightarrow I<math> such that f = hg.
2. 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)