[David P Ellerman, in his introduction,](https://forum.azimuthproject.org/discussion/2007/introduction-david-p-ellerman#latest) describes \\(\text{Het}\\)-functors as being profunctors (Personally, I like the name "\\(\text{Het}\\)-functor" rather than the uninspired, undescriptive "profunctor"),

\\[

\text{Hom}[F(X),Y] \cong \text{Het}[X,Y] \cong \text{Hom}[X,U(Y)]

\\]

So a profunctor/\\(\text{Het}\\)-functor is then like a \\(\text{Hom}\\)-functor,

\\[

\text{Hom}\_{\mathcal{C}} : \mathcal{C}^{op} \times \mathcal{C} \to \mathbf{Set}

\\]

but we're not fixed to a particular category \\(\mathcal{C}\\),

\\[

\text{Het}\_{\mathcal{D}\times\mathcal{C}} : \mathcal{D}^{op} \times \mathcal{C} \to \mathbf{Set}.

\\]

\\[

\text{Hom}[F(X),Y] \cong \text{Het}[X,Y] \cong \text{Hom}[X,U(Y)]

\\]

So a profunctor/\\(\text{Het}\\)-functor is then like a \\(\text{Hom}\\)-functor,

\\[

\text{Hom}\_{\mathcal{C}} : \mathcal{C}^{op} \times \mathcal{C} \to \mathbf{Set}

\\]

but we're not fixed to a particular category \\(\mathcal{C}\\),

\\[

\text{Het}\_{\mathcal{D}\times\mathcal{C}} : \mathcal{D}^{op} \times \mathcal{C} \to \mathbf{Set}.

\\]