# Equality of functors

I am wondering how is defined the equality of functors, seems that it is only necessary to show their equality on objects and morphisms, is it necessary to show that the natural transformation between them is the identity?

Thanks for any suggestion

• What natural transformation between them? If "two" functors have equal action on objects and arrows, what else is there to check? – Derek Elkins Mar 13 '18 at 6:25
• I can not find a definition for equality of functors, it seems to me obvious to define their equality if they are equals on objects and morphisms also, but the only relation between functors that I know is using natural transformations, so that is why I am asking for a formal definition.. – Irene Mar 13 '18 at 6:33
• Equality of functors is not something that needs to be defined; it is just equality in the same sense as any other mathematical object (i.e., set-theoretic equality, if those are your foundations). – Eric Wofsey Mar 13 '18 at 6:36
• Ok, thank you, I was confused by those natural transformations, sorry.. – Irene Mar 13 '18 at 6:37
• What may be more interesting than two functors being equal though is just if two functors are isomorphic (in the same way that it may be more interesting to ask if two objects in a category are isomorphic rather than equal). In that case you would need to produce what is called a natural isomorphism between the functors. – user171177 Mar 14 '18 at 13:00

## 1 Answer

There is no "the natural transformation between them". There is no natural transformation involved at all. If two functors are equal on all objects and equal on are morphisms, then they are equal, since a functor by definition is just a function on objects together with a function on morphisms (satisfying certain axioms).