Looking for an example of a monad that is not strong.


The reason being, a strong monad (wrt cartesian product) is an "applicative functor" (in functional programming); an example of a non-strong monad would be useful to see what's breaking in its "applicativity".



Any example of a non-strong monad?
Here is a class of examples different to Tom's: if your underlying monoidal category C is closed, then a strong monad on C is the same as a C-enriched monad, i.e. one that respects the [enrichment](https://ncatlab.org/nlab/show/enriched+category) of C given by its internal hom (this is why every monad on Set is strong, as Andrej points out). So one example would be the monad on Cat (considered as a Set-category) whose algebras are cartesian closed categories -- it is known that this is not a Cat-monad (although it does extend to Cat as a groupoid-enriched category). I would imagine that the same is true for the monad for monoidal closed categories, or in general for categories with any one sort of mixed-variance structure.


