Just to ask a question about the alternative method for showing a group/ring homomorphism $\phi$ is bijective, hence an isomorphism, by constructing an explicit inverse $f$.
Main question: Does $f$ necessarily have to be a homomorphism as well? If yes, why?
I am a bit confused about this approach since the usual approach taught in books/ lecture so far is to show injectivity by showing the kernel is trivial, etc.
Thanks for any help.