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