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

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, or extramonoidal, if anything :-) – Fosco Dec 22 '17 at 16:15
• Can you maybe expand a little the analogy you see with dinatural transformations? – Fosco Dec 22 '17 at 18:21
• @FoscoLoregian I tried to work out the details, and it didn't quite come out. The idea was to generalize from $\mathcal C^{\text{op}}\times\mathcal C\to\mathcal D$ to morphisms $A^*\otimes A\to B$ in a monoidal category with dual objects, then to generalize from there to the same thing in a bicategory rather than a monoidal category, and lastly to specialize to the case of the 2-category of monoidal categories and lax monoidal functors. The problem is that there doesn't seem to be a setting in which an oplax monoidal functor coule be regarded as the dual of a lax monoidal functor. – John Gowers Dec 22 '17 at 18:33
• Laxity is not affected by covariance type. If $F : A \to B$ is lax monoidal, and you regard $A,B$ as one-object bicategories, then $F$ will be a lax functor; if $g$ is oplax, then it will be an oplax functor; this does not affect covariance, because if $F$ reverses 1-cells it's simply a lax functor $A^\text{op}\to B$ (you "reverse" 1-cells by redefining $\otimes : (X,Y)\mapsto Y\otimes X$; of course $A^\text{op}\cong A$ if $A$ was symmetric), if it reverses 2-cells it's a lax functor $A^\text{co}\to B$ (A^co now is A^op as 1-category, I hope you get without me fixing the abuse of notation ;) ) – Fosco Dec 22 '17 at 18:41

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).
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.