Let g: A -> and f: B -> both be 1-to-1 correspondences. Prove that f o g : A->C is a 1-to-1 correspondence.

I'm studying for exam next week and trying to do a proof from sets, correspondences, induction, and an equivalence relation. I understand set proofs, now I'm doing a correspondence proof.

I was able to solve the same problem but it was proving g o f is 1-to-1. This is the pf for that.

Pf: Let g: A -> and f: B -> both be 1-to-1. Prove that f o g : A->C is a 1-to-1.

To show f is 1-to-1, we need to show that if f(x1)=f(x2), then x1=x2. Then g(f(x1))=g(f(x2)) or gf(x1) = gf(x2). This means x1 = x2 because gf is 1-to-1.

How would I change that proof to prove a correspondence?