image in an abelian category

1. Definition / Proposition

Let 20240121-image_as_limit_colimit_3038b4682e1f6d64ae501574b4029ad43f8f615e.svg be an abelian category and a morphism Then the image is defined as kernel / cokernel

20240121-image_as_limit_colimit_18f2adfabbcbb975049c6a028462e3ffdabd868e.svg

where

20240121-image_as_limit_colimit_dd4b02e90166ac8b90c82c08a3b1b7a856abd1bf.svg

2. Proof

using the image coimage isomorphism in an abelian category we get

20240121-image_as_limit_colimit_22a308e29cc5aea093a99613bbed4c59539105a7.svg

where

is an epi (cf. map into cokernel as epimorphism)
is a mono (cf. map out of kernel as monomorphism)

2.1. universal property

It suffices to show, that the coimage satisfies the universal property (cf. abelian category and coimage isomorphic to image) Suppose there exists an object 20240121-image_as_limit_colimit_26fecf54824d62034896811be8b9803e547efdf6.svg and a diagram

20240121-image_as_limit_colimit_9bc8d1249d4b9a023d6450eb936aaf8422e2f8e8.svg

Then consider

20240121-image_as_limit_colimit_de07686e57d2fcc13fc11c1550a331c7e21c3077.svg

By commutativity we conclude that

20240121-image_as_limit_colimit_f27bd8f3a6d84484b7147cd12ae9607a457d2b27.svg

is the zero morphism. As 20240121-image_as_limit_colimit_f080a51c14c1e9961d19806b7255eee52f32c5f4.svg is a mono, it follows that is already the zero morphism. Hence there exists a unique morphism

20240121-image_as_limit_colimit_9e652557d85d53fa5a23ff5d0afef49a2d037782.svg

Here uniqueness follows from the uniqueness of the induced morphism 20240121-image_as_limit_colimit_e421f32fe1072e9e156e1c64f1822421bebc9eed.svg , as any other morphism commuting with the bottom left square is solution to the universal property of