natural isomorphism and isomorphism on the objects

1. Proposition

Let 20231111-natural_isomorphism_and_isomorphism_on_the_objects_2d5d1c2bfbb1a2fca757f722875cd0787290c048.svg be categories, be functors and a natural transformation.

TFAE:

is a natural isomorphism with a canonical inverse
for it holds, that is an isomorphism in

2. Proof

2.1. 1) 2)

By assumption, there exists an 20231111-natural_isomorphism_and_isomorphism_on_the_objects_4031e48109f30e5e9653d9f269b3a02f753d36c6.svg , such that

20231111-natural_isomorphism_and_isomorphism_on_the_objects_2f591a5da8b0eafd3d8601a6faae5e1a5b27e315.svg

Analougosly for η^-1 ˆ η =&
(η^-1 ˆ η)(A) =& _(A)

2.2. 2) 1)

Let

20231111-natural_isomorphism_and_isomorphism_on_the_objects_e718bb3d9de0cd66698bba691db1ded1279a6040.svg

be the unique inverse morphism.

Then following diagram commutes,

20231111-natural_isomorphism_and_isomorphism_on_the_objects_2ca4f87786a0e716a895e1aaae4d78713509dd44.svg

hence

20231111-natural_isomorphism_and_isomorphism_on_the_objects_5a666d1f8dfedd9f03725ebcdff0cb06d16bb178.svg

Thus following diagram commutes

20231111-natural_isomorphism_and_isomorphism_on_the_objects_3fbc9b187c574c02d5a06670db863d2b1785a690.svg

hence since 20231111-natural_isomorphism_and_isomorphism_on_the_objects_e75e1ff215133a8ddf324a1e2eeff63b1569a0f6.svg is an isomorphism and thus especially also an epimorphism

20231111-natural_isomorphism_and_isomorphism_on_the_objects_d0c7b75479f04c5334f6c9306407a6d4bda10577.svg

by diagram and pasting epimorphism, it follows that 20231111-natural_isomorphism_and_isomorphism_on_the_objects_0e22c6de36922493c53b0fd8460f6244e9c1036e.svg is natural

20231111-natural_isomorphism_and_isomorphism_on_the_objects_fbd1e6c9063f27846b8796d47ab48c86901a5291.svg

note, that no choice was made, hence 20231111-natural_isomorphism_and_isomorphism_on_the_objects_aa699af8b052820cdd98bf8bee54004084151ed7.svg can be regarded as canonical.3 a