The compositions will both map to \\(f . h\\).
$$

\begin{array}{c|c}
\text{morphism }f & F~f \\\hline
\text{id}_A' & \text{id}_A \\
\text{id}_B' & \text{id}_B \\
\text{id}_C' & \text{id}_C \\
\text{id}_D' & \text{id}_D \\
f' & f \\
g' & g \\
h' & h \\
i' & i \\
f' . h' & f . h \\
g' . i' & f . h
\end{array}

$$