Recall a transformation T: X → Y is invertible if there is S: Y → X satisfying TS = idY and ST = idX. Because of the following characterization, an invertible transformation is also called a one-to-one correspondence.
T: X → Y is onto and one-to-one ⇔ There are S, S': Y → X satisfying TS = idY and S'T = idX.
The proof is then complete by showing that S and S' must be equal:
TS = idY and S'T = idX ⇒ S = idXS = S'TS = S'idY = S'.