I would like to prove that a monoidal functor \Phi\colon \mathbf{V}\to \mathbf{V'} induces a functor \Phi^\#\colon \mathbf{V}\text{-Cat}\to \mathbf{V'}\text{-Cat}
and in particular I would like to check that if $\Phi$ admits a left/right adjoint, then $\Phi^\#$ admits one too. The problem is that I'm stuck in finding the "right" definitions involved (especially that of monoidal functor); I have to silently suppose that $\Phi$ is strong (or "non-lax") monoidal, i.e. \Phi(A\otimes B)\cong \Phi(A)\otimes '\Phi(B), \Phi(I)\cong I' for all $A,B\in \mathbf{V}$ and the "initial" objects $I\in \mathbf V$, I'\in \mathbf V'. Such a restrictive assumption leaves me unsatisfied, but I'm not really keen on non-strict monoidal functors...
As a side question, it seems to me this is a well-established result in enriched category theory, but I'm not able to find a precise reference proving the result from the beginning: Kelly treats the result as a well known folklore, saying in the first pages of Basic concepts of ECT
[we do not] discuss the change of base-category given by a symmetric monoidal functor \mathbf{V}\to \mathbf{V'} and the induced 2-functor \mathbf{V}\text{-Cat}\to \mathbf{V'}\text{-Cat}, [...]
and John Gray, in his article Closed Categories, Lax Limits and Homotopy Limits just gives a statement of the claim I would like to prove. Again, can you help me?
Thanks a lot.