Applicative functors are well-known and well-loved among Haskellers, for their ability to apply functions in an effectful context.
In category-theoretic terms, it can be shown that the methods of Applicative:
are equivalent to having a Functor f with the operations:
the idea being that to write pure you just replace the () in unit with the given value, and to write (<*>) you squish the function and argument into a tuple and then map a suitable application function over it.
Moreover, this correspondence turns the Applicative laws into natural monoidal-ish laws about unit and (**), so in fact an applicative functor is precisely what a category theorist would call a lax monoidal functor (lax because (**) is merely a natural transformation and not an isomorphism).
Okay, fine, great. This much is well-known. But that's only one family of lax monoidal functors – those respecting the monoidal structure of the product. A lax monoidal functor involves two choices of monoidal structure, in the source and destination: here's what you get if you turn product into sum:
It seems like turning sum into other monoidal structures is made less interesting by unit :: Void -> f Void being uniquely determined, so you really have more of a semigroup going on. But still:
Are other lax monoidal functors like the above studied or useful?
Is there a neat alternative presentation for them like the Applicative one?
I'm somewhat expecting the answer "no, because no", but you never know until you ask :)