I am reading Masumura, Commutative algebra, Chapter 10: Derivation. The following is in pages 177, 178.
Two extensions $(C, \varepsilon, i)$ and $(C_1, \varepsilon_1, i_1)$ are said to be isomorphic if there exists a ring homomorphism $f: C \to C_1$ such that $\varepsilon_1 f= \varepsilon$ and $fi =i_1$ [it means the diagram commutes]. Such $f$ is necessarily unique.
Question: Why is $f$ unique?
Thanks!