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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

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  • $\begingroup$ What natural transformation between them? If "two" functors have equal action on objects and arrows, what else is there to check? $\endgroup$ Commented Mar 13, 2018 at 6:25
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    $\begingroup$ 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.. $\endgroup$
    – Irene
    Commented Mar 13, 2018 at 6:33
  • $\begingroup$ 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). $\endgroup$ Commented Mar 13, 2018 at 6:36
  • $\begingroup$ Ok, thank you, I was confused by those natural transformations, sorry.. $\endgroup$
    – Irene
    Commented Mar 13, 2018 at 6:37
  • $\begingroup$ 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. $\endgroup$
    – user171177
    Commented Mar 14, 2018 at 13:00

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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 all 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).

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  • $\begingroup$ How are two objects equal? A natural isomorphism forall x, F x <-> G x . $\endgroup$ Commented Apr 8, 2021 at 22:43
  • $\begingroup$ I don't know what you're trying to say. $\endgroup$ Commented Apr 8, 2021 at 22:49

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