Rather, the identity map is a homomorphism.

This makes sense.

You should start proving this statement as you would any other universal statement. Namely, one proves "for all x, P(x)" by fixing an arbitrary x and showing P(x). Here is such a universal statement. By this I don't mean that is a string of characters that starts with ; rather, I mean that the whole claim , when you unfold some definitions, starts with the words "for all."