So I'm reading book Conceptual Mathematics, 2nd Edition A first introduction to categories. Several times I was struggling with exercises related to proofs because I feel like authors are asking me to prove something so obvious that it doesn't make sense to prove prove it at all. E.g below authors are using associativity and substitution to deduce needed equation in the category of endomaps.
But I can't understand why we can't just say that identities don't change the value of a composition so we are free to add them or remove as we want, therefore any manipulation with composition that affects only identities is valid and this particular too. That way we can avoid using this chain of substitutions that would feel even more redundant if we were facing exercise where 100s of functions are composed.
My question is - am I missing something here? Because this a very common for me in many textbooks - I'm asked how to prove that 2*2 is 4 and I don't understand how to start because it feels so obvious yet I need to present some proof. How do I learn to proof "obvious" things? I'm not saying that I'm in any way superior - on the contrary I'm often surprised that there is a way and need to make a proof. How do I judge if I have proven something or not?