$$ \def\cat#1{{\mathbf{#1}}} \def\opcat#1{{\mathbf{#1}^{\mathrm{op}}}} $$ A symmetric lax monoidal functor $F : \cat{C} \rightarrow \cat{D}$ between monoidal categories $(\cat{C}, \otimes, I)$, $(\cat{D}, \oplus, J)$ is a functor equipped with coherence maps:

$$ \phi_{A,B} : F A \oplus F B \rightarrow_{\cat{D}} F (A \otimes B) $$


$$ \phi : J \rightarrow_{\cat{D}} F I $$

such that an associativity diagram, two unit diagrams, and a symmetry diagram involving the natural isomorphisms of the monoidal structures commute.

It turns out these four diagrams correspond precisely to the diagrams of a commutative monoid object in a particular symmetric monoidal category. More precisely, given a self-enriched monoidal category $(\cat{D}, \oplus, J)$, and a $\cat{D}$-enriched monoidal category $(\cat{C}, \otimes, I)$, the functor category $[\cat{C}, \cat{D}]$ inherits a symmetric monoidal structure known as Day convolution.

The tensor $\oplus_{\mathrm{Day}} : [\cat{C}, \cat{D}] \times [\cat{C}, \cat{D}] \rightarrow [\cat{C}, \cat{D}]$ is given by:

$$ F \oplus_{\mathrm{Day}} G = c \mapsto \int\limits^{a, b \in \cat{C}} (a \otimes b \rightarrow_{\cat{C}} c) \oplus F a \oplus G b $$

and the unit by a functor $\Delta I : [\cat{C}, \cat{D}]$ that is naturally isomorphic to $I \rightarrow_{\cat{C}} -$.

If you convince yourself that this really does form a symmetric monoidal structure on the aforementioned $\cat{D}$-functor category, then it is not much more work to conclude that to be a (symmetric) lax monoidal functor from $\cat{C}$ to $\cat{D}$ is to be a (commutative) monoid in this category, and vice versa.

So far so good. The problem is not all lax monoidal functors occur in a functor category where the domain category is enriched in the codomain category.

Consider for example the phenomenon of an "oplax" monoidal functor. If we take the definitions of $\phi_{A, B}$ and $\phi$ above and reverse the arrows, the existence of these opposite laxities describes an oplax monoidal functor. We can define an oplax monoidal functor without resorting to a new concept by defining it as the opposite functor of a standard lax monoidal functor. In other words, a functor $F : \cat{C} \rightarrow \cat{D}$ is oplax monoidal iff its opposite functor $F^{\mathrm{op}} : \opcat{C} \rightarrow \opcat{D}$ is lax monoidal.

Given a functor category $[\cat{C}, \cat{D}]$ suited for the Day convolution monoidal structure, the opposite category $[\opcat{C}, \opcat{D}]$ seems by definition unsuited to a similar monoidal structure. If I understand correctly both $\opcat{C}$ and $\opcat{D}$ are enriched in $\cat{D}$ (their existence relying on its symmetry as a monoidal category). Since neither is enriched in $\opcat{D}$, we can't form the Day convolution monoidal structure discussed above.

Is there some way we can equip this "opposite functor category" $[\opcat{C}, \opcat{D}]$ with a monoidal structure as well, such that we can recognize oplax monoidal functors as monoids/comonoids?

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    $\begingroup$ I don't know why you speak of enrichment. What one needs is the existence of colimits in $\mathcal{D}$ (even very large ones when $\mathcal{C}$ is large). In the formula for the Day convolution (which is just a good-old left Kan extension, by the way) take the copower of the set $\mathrm{Hom}(a \otimes b,c)$ with $F(a) \oplus G(b)$. For oplax functors, take the dual concept defined by ends instead of coends, and consider comonoids instead of monoids. $\endgroup$ Feb 5 '20 at 18:10
  • $\begingroup$ (Remark: I am not a fan of posting answers in comments. But my comment asks for a clarification of the assumptions made in the post. When they are clear, I can post an answer as well.) $\endgroup$ Feb 5 '20 at 18:13
  • $\begingroup$ @MartinBrandenburg The reason I speak of enrichment is because I learnt my Day convolution from Definition 2.1 of this page: ncatlab.org/nlab/show/Day+convolution. I wasn't aware the formula worked with a weaker constraint than that. The specific category I'm actually interested in is the one of sets under various tensors, so if the construction you're proposing works for modeling lax monoidal functors $\mathbf{Set}^{\mathrm{op}} \rightarrow \mathbf{Set}^{\mathrm{op}}$ as monoid objects it's exactly what I'm looking for. $\endgroup$ Feb 5 '20 at 19:26
  • $\begingroup$ In my (likely imperfect) understanding, the enrichment of $C$ in $D$ is significant because we want the hom $a \otimes b \rightarrow_C c$ to be an object of $D$, so that we can tensor it with $F a$ and $F b$. I only have a very superficial understanding of the concept, so maybe it's obvious how one can get by without this. $\endgroup$ Feb 5 '20 at 19:52
  • $\begingroup$ Well as I said, when you have a category with coproducts (or just copowers), $S \otimes X$ is defined for every set $S$ and every object $X$. Now take $S = \mathrm{Hom}(a \otimes b,c)$. You can also consult Day's original article or other resources on the topic. I also cover Day convolution in my thesis, Section 5.1.2. $\endgroup$ Feb 5 '20 at 22:18

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