Monoidal functor

Summary

A monoidal functor is a functor between monoidal categories that is compatible with the tensor products and units. It comes with maps $F(A) \otimes F(B) \to F(A \otimes B)$ and $I \to F(I)$ that respect the associators and unitors. When these maps are isomorphisms the functor is called strong, and when they are identities it is called strict. Monoidal functors send monoid objects to monoid objects, and they are the morphisms of the category of monoidal categories.