$$ \def\cat#1{{\mathbf{#1}}} \def\opcat#1{{\mathbf{#1}^{\mathrm{op}}}} $$ 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:

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

and

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

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?

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$1more comment