Consider \(\mathcal{C}=\boxed{\bullet \rightarrow \bullet}\) and \(\mathcal{D}=\boxed{\bullet \rightrightarrows \bullet}\). Give two functors that act the same on objects but differently on morphisms.
Let the two functors map the left object to the left object and right object to the right object. The first functor maps the nontrivial morphism to the upper morphism in \(\mathcal{D}\), whereas the second maps it to the lower morphism.