In category theory, a strong monad is a monad on a monoidal category with an additional natural transformation, called the strength, which governs how the monad interacts with the monoidal product.
Strong monads play an important role in theoretical computer science where they are used to model computation with side effects[1].
A (left) strong monad is a monad (T, η, μ) over a monoidal category (C, ⊗, I) together with a natural transformation tA,B : A ⊗ TB → T(A ⊗ B), called (tensorial) left strength, such that the diagrams commute for every object A, B and C. For every strong monad T on a symmetric monoidal category, a right strength natural transformation can be defined by
The Kleisli category of a commutative monad is symmetric monoidal in a canonical way, see corollary 7 in Guitart[2] and corollary 4.3 in Power & Robison[3].
One interesting fact about commutative strong monads is that they are "the same as" symmetric monoidal monads.
[4] More explicitly, and the conversion between one and the other presentation is bijective.