In category theory, a strong monad over a monoidal category is a monad together with a natural transformation , called (tensorial) strength, such that the diagrams
and
commute for every object , and .
For every strong monad T on a symmetric monoidal category, a costrength natural transformation can be defined by
A strong monad T is said to be commutative when the diagram
commutes for all objects and .
One interesting fact about commutative strong monads is that they are "the same as" symmetric monoidal monads. More explicitly,
and the conversion between one and the other presentation is bijective.