A (strong) monoidal functor $F:C\to C'$ between monoidal categories is a functor equipped with natural isomorphisms $$ F_0:I'\cong F(I) $$ $$ (F_2)_{A,B}:F(A)\otimes'F(B)\cong F(A\otimes B) $$ such that some certain diagrams commute (see https://ncatlab.org/nlab/show/monoidal+functor).
A (right-)closed monoidal category is a monoidal category where for every object $B$, the functor $-\otimes B$ has a right adjoint $[B,-]$. (Note we can assemble the $[B,-]$ functors into a single bifunctor $[-,-]:C^{op}\times C\to C$ in a canonical way).
I was wondering if someone could provide a precise definition of what it means to be a "closed monoidal functor", if such a definition exists?
The best I could find was a discussion on how lax monoidal functors give rise to lax closed functors on https://ncatlab.org/nlab/show/closed+functor.