Old - Promonoidal category
A promonoidal category is a category $π$ together with profunctors $$π(\bullet,\bullet;\bullet) \colon π^{op} Γ π^{op} Γ π \to \mathbf{Set}, \mbox{ and } π(\bullet) \colon π \to \mathbf{Set},$$ and natural coherence isomorphisms $$Ξ± \colon π(\bullet ;\bullet_1 \otimes \bullet) Γ π(\bullet_1 ;\bullet \otimes \bullet) β π(\bullet ;\bullet \otimes \bullet_2) Γ π(\bullet_2 ;\bullet \otimes \bullet),$$
$$\lambda \colon π(\bullet ;\bullet_1 \otimes \bullet) Γ π(\bullet_1) \cong π(\bullet; \bullet).$$
$$\rho \colon π(\bullet ;\bullet \otimes \bullet_2) Γ π(\bullet_2) \cong π(\bullet; \bullet),$$ $$\begin{aligned} \alpha \colon & \left( \int^{A \in π} π(X,Y;A) Γ π(A,Z;B) \right) \to \left( \int^{A \in π} π(Y,Z;A) Γ π(X,A;B) \right), \\ \lambda \colon & \left( \int^{A \in π} π(A) Γ π(A,X;Y) \right) \to π(X,Y), \\ \rho \colon & \left( \int^{A \in π} π(A) Γ π(X,A;Y) \right) \to π(X,Y). \end{aligned}$$ satisfying the pentagon and triangle equations: $Ξ± β¨Ύ Ξ± = (Ξ±β1)β¨ΎΞ±β¨Ύ(1βΞ±)$, and $(Ο β 1) = Ξ± β¨Ύ (Ξ» β 1)$. $$\rho(a \mid v) > w =\\ let\\ \alpha(v \mid w) \to (w’ \mid v’)\\ in\\ \lambda(a \mid w’) > v’,$$ $$\left{ \begin{aligned} &let\\ \alpha(u \mid v) \to (v’ \mid u’)\\ in\\ \\ &let\\ \alpha(u’ \mid w) \to (w’ \mid u’’)\\ in\\ \\ &let\\ \alpha(v’ \mid w’) \to (w’’ \mid v’)\\ in\\ \\ &(w’’ \mid v’’ \mid u’’) \end{aligned} \right} = \left{ \begin{aligned} &let\\ \alpha(v \mid w) \to (w’ \mid v’)\\ in\\ \\ &let\\ \alpha(u \mid v’) \to (v’’ \mid u’)\\ in\\ \\ &(w’ \mid v’’ \mid u’) \end{aligned} \right}.$$