Let $\mathcal C,\mathcal D,\mathcal E$ be monoidal categories, let $g$ be an oplax monoidal functor from $\mathcal C$ to $\mathcal D$ and let $G$ be a lax monoidal functor from $\mathcal D$ to $\mathcal E$. 


Let $F$ be a lax monoidal functor from $\mathcal C$ to $\mathcal E$. Now let $\phi$ be a natural transformation from $F$ to $Gg$ making the following diagrams commute for any objects $x,y$ of $\mathcal C$:


$$
\require{AMScd}
\begin{CD}
F(x)\otimes F(y) @>{\phi\_x\otimes\phi\_y}>> Gg(x)\otimes Gg(y)\\
@V{m\_F}VV @VV{m\_G}V \\
 @. G(g(x)\otimes g(y))\\
@V{m\_F}VV @AA{G(m\_g)}A \\
F(x\otimes y) @>>{\phi\_{x\otimes y}}> Gg(x \otimes y)
\end{CD}
$$


$$
\begin{CD}
I\_{\mathcal E} @>{\epsilon\_G}>> G(I\_{\mathcal D}) \\
@V{\epsilon\_F}VV @VV{G(\epsilon\_g)}V \\
F(I\_{\mathcal C}) @>>{\phi\_{I\_{\mathcal C}}}> Gg(I\_{\mathcal C})
\end{CD}
$$


Is there a name for such natural transformations? I'm tempted to call them *bimonoidal natural transformations*, since they seem to be analogous in some sense to the bifunctors $\mathcal C^{\text{op}}\times\mathcal C\to\mathcal D$ that are used in the definition of (co)ends. 


*Note:* A special case of this is when $F$ is in fact strong monoidal. In that case, this is the 'obvious' idea of a monoidal natural transformation from a lax monoidal functor to an oplax monoidal functor.



What is the name for a natural transformation that has both lax and oplax monoidal properties?
I don't know a good name for such transformations, but I can give you a reference and some more information about them. First of all note that more generally you can define a monoidal transformation $f F \to G g$ when $F$ and $G$ are lax monoidal and $f$ and $g$ are colax monoidal. The axioms then become hexagons (with one side degenerate in the unit case).


Such transformations may seem *ad hoc*, but they actually arise very naturally. They are the 2-cells in a *double category* whose objects are monoidal categories and whose two kinds of morphism are lax and colax monoidal functors. This double category was first defined in section 2.2 of [Adjoint for double categories](http://www.numdam.org/item/CTGDC_2004__45_3_193_0/) by Grandis and Paré, as a special case of an analogous double category whose objects are *double* categories.


In [Comparing composites of left and right derived functors](https://arxiv.org/abs/0706.2868) (example 4.8) I observed that an analogous double category can be defined for the algebras over any 2-monad, in which case the corresponding axiom on the 2-cells is that a certain *cube* commutes (the six sides of the hexagons corresponding to the six faces of the cube).


In fact this construction even sits inside something with a universal property: given a 2-category $K$, let $Q(K)$ denote the *triple* category of "quintets", whose objects are those of $K$, whose three kinds of morphisms are all those of $K$, whose three kinds of 2-cells are all 2-cells in $K$ ("quintets"), and whose 3-cells are commutative cubes in $K$. Any 2-monad on $K$ induces a "triple monad" $Q(T)$ on $Q(K)$, i.e. a monad in the 2-category of triple categories, triple functors, and "transversal" transformations. The triple category of algebras for $Q(T)$ (i.e. its Eilenberg-Moore object in this 2-category of triple categories) has as its objects $T$-algebras and as its three kinds of morphisms the strict, lax, and colax $T$-morphisms, while its 2-cells relating lax and colax $T$-morphisms are precisely these.


