In an abelian category, let $$0\longrightarrow A \overset{f}{\longrightarrow}B\overset{g}{\longrightarrow}C\longrightarrow0$$ be a split short exact sequence with $\ell f=1_A,gr=1_C$. Is the sequence $$0\longrightarrow C \overset{r}{\longrightarrow}B\overset{\ell}{\longrightarrow}A\longrightarrow0$$ also short exact? How can I prove this? Is this also true in $\mathsf{Grp}$?
Lastly, I proved $\operatorname{Im}f\cong A$, but didn't manage to prove $\operatorname{Ker}\ell \cong C$. I'd appreciate some help!