The textbook I'm using says that a morphism might have only a section, or only a retraction, but I can't work out an example.
Take objects $A$ and $B$, and morphisms $f$ and $g$,
nice diagram: $\hskip1in$
ugly diagram: $\matrix{B \hspace{-0.11in} & & \hspace{-0.1in}\xrightarrow{\quad\text{Id}_B\quad} & &\hspace{-0.1in} B\\ & g\searrow& \hspace{-1in}& \nearrow f\\ & \text{}\hskip{-1in}& A &}$
So according to my book, $g$ is a section for $f$ because $g;f =\text{Id}_B$
But that can be rearranged to this just be removing the $\text{Id}_B$ and adding an $\text{Id}_A$:
nice diagram: $\hskip1in$
ugly diagram: $\matrix{A \hspace{-0.11in} & & \hspace{-0.1in}\xrightarrow{\quad\text{Id}_A\quad} & &\hspace{-0.1in} A\\ & f\searrow& \hspace{-1in}& \nearrow g\\ & \text{}\hskip{-1in}& B &}$
So $g$ is a retraction for $f$ because $f;g = \text{Id}_A$.
It seems to me that this would apply everywhere, and so any morphism that has a section must also have a retraction. What am I missing?