What is the name for a natural transformation that has both lax and oplax monoidal properties?

Question

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.

Answer 1

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 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 (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.