Let $C\_1$ and $C\_2$ be monoidal categories (not necessarily symmetric or strict) and let $\Psi : C\_1 \rightarrow C\_2$ be a strong monoidal functor. Is it possibly to construct a strict monoidal functor $\Psi'$ which is naturally isomorphic to $\Psi$? If not, is this possible if we assume that the $C\_i$ are strict?



Strictifying strong monoidal functors
Take $C\_2$ to be a non-strict monoidal category, $C\_1$ to be its strictification and $\Psi$ to be the equivalence. Since $C\_2$ is not strict there's no strict monoidal equivalence between $C\_1$ and $C\_2$, so in particular $\Psi$ is not naturally isomorphic to a strict functor.


