maybe, as suggested by Simon, that your approach gives some hints on the completeness of $\bf Cat'$? (see the comments above)

I'm wondering which are the completeness properties of $\bf Cat'$. For instance, there seems to be no terminal object; in fact, in $\bf Cat'$ there are two morphisms to the terminal category: one covariant and one contravariant. And the same problem holds for the initial object.

@martti : thanks for the very useful references. In fact I had a similar idea of proof based on factorization. So it seems that it is true for instance for $C = Set_f^G$ for any finite category $G$ (as well as for their dual).