Let $f: M\to N$ be an $A$-module homomorphism . Establish 1-1 correspondence between these 2 sets :
$$\{\phi: K \to \text{Ker}(f) \mid \phi \text{ is an isomorphism}\}$$ and $$\{ g: K\to M \mid 0 \to K \to M \to N \text{ exact} \}$$
How to define the isomorphisms ?