In short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$, prove $f: A \to B$ is a kernel of $g$

Question

Let $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ be a short exact sequence in an abelian category. I want to prove $f: A \to B$ is a kernel of $g$. I am really struggling to even see where to start.

Answer 1

The Freyd Mitchell embedding theorem says we may think of this short exact sequence as a short exact sequence of $R$-modules, and thus we can work element-wise instead of using categorical definitions.

Since the short exact sequence is exact at $A$, $ker(f)=im(0\to A)=0$, so $f$ is injective. Since the short exact sequence is exact at $B$, $ker(g)=im(f)=A\subseteq B$, conflating $A$ with $f(A)$ since $f$ is injective. Whence the result.