Strong monad

In category theory, a strong monad over a monoidal category ("C", ⊗, I) is a monad ("T", η, μ) together with a natural transformation "t""A,B" : "A" ⊗ "TB" → "T"("A" ⊗ "B"), called ("tensorial") "strength", such that the diagrams
commute for every object "A", "B" and "C" (see Definition 3.2 in ).
If the monoidal category ("C", ⊗, I) is closed then a strong monad is the same thing as a "C"-enriched monad.
Commutative strong monads.
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 formula_2 and formula_3.
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.