Let $C$ and $D$ be monoidal categories. Let $T:C \to D$ be a functor that gives the equivalence of $C$ and $D$ as just categories.
My question is whether such an equivalence $T$ is always a monoidal equivalence or not.
If this is true, could you give me a proof?
Thank you.
Certainly not, for roughly the same reason that if $X,Y$ are monoids, then a bijection between the underlying sets need not respect the monoid operations. If you understand this claim, you can answer your question too.