Problems with proofs of Robert Geroch mathematical physics

[impblack]

Hello guys, I'm new in this forum, this is my first Thread.

I've started reading Robert Geroch's Mathematical Physics recently and I've been having problems with some of the proofs that involve monomorphism.

He defines monomorphism the following way (pg 4):
let ψ be a morphism between A and B. For any object X, let α and α' be morphism form X to A such that ψoα=ψoα', then, if ψ is a monomorphism, α=α'.

But then in some proofs later on, when he wants to demonstrate that some mappings are monomorphism he uses specific cases for X, the simplest cases he can find. But i was thinking that it would only be a valid proof if the definition of monomorphims was: There is at least one object X and not For any object X.

A proof for example (pg 5):
https://imagizer.imageshack.us/v2/706x397q90/538/S51cvZ.jpg

Where he uses a specific X (a set with only one member). Is there no loss of generality?

 

[Cruikshank]

I believe there is no loss of generality. There are three sets, X, A and B. We are given A and B, and that phi is a monomorphism from A to B.
X is introduced as a tool to show a = a'. If it is true for X, it is true for any other X, because a and a' are elements of A, not X. Whether a = a'
or not is already fixed by A B and phi. The simplest analog I can think of to this kind of reasoning is finding the coefficients in a partial fractions
decomposition: you can use any values of x you want to get a system of equations for the coefficients, but once you get those coefficients, it
doesn't matter what values you used. I hope that helps.