In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules.

A profunctor (also named distributor by the French school and module by the Sydney school) ${\displaystyle \,\phi }$ from a category ${\displaystyle C}$ to a category ${\displaystyle D}$, written

is defined to be a functor

where ${\displaystyle D^{\mathrm {op} }}$ denotes the opposite category of ${\displaystyle D}$ and ${\displaystyle \mathbf {Set} }$ denotes the category of sets. Given morphisms ${\displaystyle f:d\to d',g:c\to c'}$ respectively in ${\displaystyle D,C}$ and an element ${\displaystyle x\in \phi (d',c)}$, we write ${\displaystyle xf\in \phi (d,c),gx\in \phi (d',c')}$ to denote the actions.

Using that the category of small categories ${\displaystyle \mathbf {Cat} }$ is cartesian closed, the profunctor ${\displaystyle \phi }$ can be seen as a functor

where ${\displaystyle {\hat {D}}}$ denotes the category ${\displaystyle \mathrm {Set} ^{D^{\mathrm {op} }}}$ of presheaves over ${\displaystyle D}$.

A correspondence from ${\displaystyle C}$ to ${\displaystyle D}$ is a profunctor ${\displaystyle D\nrightarrow C}$.

An equivalent definition of a profunctor ${\displaystyle \phi :C\nrightarrow D}$ is a category whose objects are the disjoint union of the objects of ${\displaystyle C}$ and the objects of ${\displaystyle D}$, and whose morphisms are the morphisms of ${\displaystyle C}$ and the morphisms of ${\displaystyle D}$, plus zero or more additional morphisms from objects of ${\displaystyle D}$ to objects of ${\displaystyle C}$. The sets in the formal definition above are the hom-sets between objects of ${\displaystyle D}$ and objects of ${\displaystyle C}$. (These are also known as het-sets, since the corresponding morphisms can be called heteromorphisms.) The previous definition can be recovered by the restriction of the hom-functor ${\displaystyle \phi ^{\text{op}}\times \phi \to \mathbf {Set} }$ to ${\displaystyle D^{\text{op}}\times C}$.