induced morphism between colimits and functor

1. Proposition

Let 20240207-map_between_colimits_and_functor_2d5d1c2bfbb1a2fca757f722875cd0787290c048.svg be categories and 20240207-map_between_colimits_and_functor_570624dc5b21d1f0a192cc0f7dad2a4f870badf0.svg be a functor. Suppose 20240207-map_between_colimits_and_functor_f04e095e75ffa5249d71a7e72a48e08b90f434a1.svg is a diagram and

20240207-map_between_colimits_and_functor_db86141f9ac93de69359bb3252b2c30fd4814909.svg

the colimit.

Then there exists a natural morphism

20240207-map_between_colimits_and_functor_11fff92c6fbeece59687c2501c16fa1674499fa8.svg

2. Proof

2.1. existence

Note that

20240207-map_between_colimits_and_functor_6f4389e77fc09c49c226238212115dfdf436b8ae.svg

commutes, hence

20240207-map_between_colimits_and_functor_f77b9b8ddc187753582d138688aabdd06c5d825d.svg

2.2. naturality

follows from

Date: nil

Author: Anton Zakrewski

Created: 2024-10-14 Mo 08:54