Let $C_1,C_2$ be monoidal categories (aka tensor categories) with tensor bifunctor $$\otimes_i: C_i\times C_i\to C_i$$ and tensor units $1_i$.
Assume I have a monoidal functor $F:C_1\to C_2$, it comes with an isomorphism $F(1_1)\simeq 1_2$ and a natural family $$F(U,V):F(U\otimes_1 V)\to F(U)\otimes_2 F(V).$$
Further, assume that $G:C_2\to C_2$ is a monoidal equivalence of $C_2$.
The map $F(U,V)\otimes_2 id:F(U\otimes_1 V)\otimes_2 F(W)\to (F(U)\otimes_2 F(V))\otimes_2 F(W)$ is a morphism in $C_2$ and thus I can look at $G(F(U,V)\otimes_2 id)$.
Is the last map equal to the map $G(F(U,V))\otimes_2 G(id)=G(F(U,V))\otimes_2 id$?