A promonoidal category is like a monoidal category in whose structure (namely, tensor product and unit object) we have replaced functors by profunctors. It is a categorification of the idea of a boolean algebra.
A promonoidal category is a pseudomonoid in the monoidal bicategory Prof. This means that it is a category $A$ together with
Recalling that a profunctor $A ⇸ B$ is defined to be a functor $B^{op}\times A \to Set$, we can make this more explicit. We can also generalize it by replacing $Set$ by a Benabou cosmos $V$ and $A$ by a $V$-enriched category; then a profunctor is a $V$-functor $B^{op}\times A \to V$.
Thus, we obtain the following as an explicit definition of promonoidal $V$-category: we have the following data
A $V$-category $A$.
A $3$-ary functor $P:A^\op \otimes A \otimes A\to V$. For notational clarity, we may write $P(a,b,c)$ as $P(a,b \diamond c)$.
A $V$-functor $J:A^{op}\to V$.
and natural isomorphisms
$\lambda_{ab}:\int^x (J(x) \otimes P(b,a \diamond x))\to A(b,a)$
$\rho_{ab}: \int^x ( J(x)\otimes P(b,x \diamond a))\to A(b,a)$
$\alpha_{abcd}: \int^x (P(x,a\diamond b)\otimes P(d,x\diamond c)) \to \int^x(P(x,b\diamond c)\otimes P(d,a\diamond x))$
satisfying the pentagon and unit axioms.
Since any functor induces a representable profunctor, any monoidal category can be regarded as a promonoidal category. A given promonoidal category arises in this way if and only if the profunctors $P$ and $J$ are representable.
A promonoidal structure on $A$ suffices to induce a monoidal structure on $V^{A^{op}}$ by Day convolution. In fact, given a small $V$-category $A$, there is an equivalence of categories between
the category of pro-monoidal structures on $A$, with strong pro-monoidal functors between them, and
the category of biclosed monoidal structures on $V^{A^{op}}$, with strong monoidal functors between them.
A promonoidal structure on $A$ can be identified with a particular sort of multicategory structure on $A^{op}$, i.e. with a co-multicategory structure on $A$. The set $P(x, y, z)$ is regarded as the set of co-multimorphisms $x \to (y,z)$.
More generally, we define a co-multicategory $\bar A$ as follows. The objects of $\bar A$ are the objects of $A$. The co-multimorphisms $b\to a_1\dots a_n$ in $\bar A$ are defined by induction on $n$ as follows: $\bar A(b;)=Jb$, and $\bar A(b;a_1,\dots,a_{n+1})=\int^x\bar A(x;a_1,\dots,a_n)\otimes P(b,x\diamond a_{n+1})$.
Not every co-multicategory arises from a promonoidal one in this way. Roughly, a promonoidal category is a co-multicategory whose $n$-ary co-multimorphisms are determined by the binary, unary, and nullary morphisms. In general, co-multicategories can be identified with a certain sort of “lax promonidal category”.
Brian Day introduced the notion of a “premonoidal” category in (Day 1970), and later renamed this to a “promonoidal” category in (Day 1974) while reformulating the identity and associativity isomorphisms $\lambda,\rho,\alpha$ explicitly in terms of profunctor composition. However, note that his definition is op’d from the definition used in this article, in the sense that a Day-promonoidal structure on a category $C$ corresponds to a pseudomonoid structure on $C^{op}$ in Prof. In particular, one example Day considers is that of a closed category, which is actually a co-promonoidal category in the sense used here (analogous to the co-promonoidal structure on a multicategory described above).
Brian Day, On closed categories of functors, Lecture Notes in Mathematics 137 (1970), 1-38.
Brian Day, An embedding theorem for closed categories, Lecture Notes in Mathematics 420 (1974), 55-64.
Day, Panchadcharam and Street, On centres and lax centres for promonoidal categories.