@WillieWong the example you given is very interesting. it is an example of affirmative answer to my post. in this example $\phi$ is a diffeomorphism and Gauss maps are also diffeomorphism so the map $\psi$ is detrtmined uniquely. so I wonder if thete are some complicated example for example degrere of $\phi$ is 2 or >2? for Riemann surfaces embeded on space with holomorphic Gauss maps can we provid a holomorphic $\psi$ when $\phi $ is holomorphic?

@WillieWong Yes I see thanks.

@WillieWong Can one find a possible reation between these two questions?

@MarkGrant so I realize that maybe the abelian case is not known

@MarkGrant I meant $G\mapsto G'=[G,G]$ yes it does not vanish on free group. Now I see why did you pointed out to abelian group. thanks again for your attention to my question.

@MarkGrant I did not see your two previous comment. the reason I considered the category of non abelian group is thst I was initially interested in commutator functor and wss curiious if it is derived functor. so i considered non abelian category. Any way what is the answer for abelian group?

@MarkGrant thank you very much for your very helpful comment

@BugsBunny I would appreciate if you read my previous two comments.

@YCor what is the difference between endofunctor and a functor from a category to itself? You are right. I was mistaken touse the terminilogy for non abelian category. But what are obstructions to have the derived functors in non abelian categories? In what step of construction of derived functor we need "abelian"-ness?

@MarkGrant I was not aware of this in abelian category. Is the proof obvious in this case?