This method of quotienting by the kernel in order to fix injectivity is much more enlightening than the normal proof of defining psi first, and just showing that it is well defined and an isomorphism. Thanks!
Nice! I think this video gave me a insight: the procedure to find the isomorphism is like when we have a function and we want a bijection out of it. The case of the function say F:A -> B, we can simply restrict it to its image, to turn it surjective, and group together the elements of the inverse image of each element of Im F, to turn it injective. This give us a bijection F: Disjunct union of the inverse image of each element of Im F -> Im F. What happens in group theory seems to be just a abstraction of this procedure, and the relations we have on the kernel give us a more natural way to group together the elements of the domain of the function. And I guess this procedure is done for other algebraic structures the same way.
