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.

dimonoidal, orextramonoidal, if anything :-) $\endgroup$ – Fosco Dec 22 '17 at 16:15dualof a lax monoidal functor. $\endgroup$ – John Gowers Dec 22 '17 at 18:33