Definition
A prostrong monad over a multicategory - promonoidal category is a monad endowed with two strengths
st_r﹕ 𝕍(X,Y;Z) → 𝕍(X,TY;TZ).$$ These must satisfy the following axioms. 1. Unitality, $T(λ(u|v)) = λ(u|st_l(v))$ and $T(ρ(u \mid v)) = ρ(u \mid st_r(v))$. 2. Associativity,$$α(st_l(v) \mid st_l(w)) = \left\{ \begin{aligned} & let\ α(v|w) → (w' \mid v')\ in\ \\ & (w' \mid st_l(v')) \end{aligned} \right\};$$ 3. Neutrality, $$η > st_l(p) = p < η\mbox{ and }η > st_r(p) = p < η\ ;$$ 4. Multiplicativity, $$μ > st_l(p) = st_l(st_l(p)) < μ\mbox{ and }μ_r > st_r(p) = st_r(st_r(p)) < μ\ ;$$ 5. Coherence,$$α(st_r(v)\mid st_l(w)) = \left\{\begin{aligned} & let\ α(v \mid w) → (w' \mid v')\ in \\ & (st_r(w') \mid st_l(v')) \end{aligned}\right\}.$$ ## References - Personal communication, [James Hefford](James%20Hefford). - [Note on Monoidal Monads (Day, 1976)](private-notes/Note%20on%20Monoidal%20Monads%20(Day,%201976).md). Day provides a definition of promonoidal promonad, a prostrong monad is its weakened version.