> So then, do profunctors/ hets/modules satisfy an analogous composition law to hom-functors?
>
> \[ \text{Het}(f,g) \circ \text{Het}(j,k) \to \text{Het}(f \circ j, k \circ g) \]

Wikipedia states how to [compose two profunctors](https://en.wikipedia.org/wiki/Profunctor#Composition_of_profunctors):

>Using the [cartesian closure](https://en.wikipedia.org/wiki/Cartesian_closed_category) of \\(\mathbf{Cat}\\), the [category of small categories](https://en.wikipedia.org/wiki/Category_of_small_categories), the profunctor \\(\phi\\) can be seen as a functor
>
> \[\hat{\phi} \colon \mathcal{C}\to\mathrm{Set}^{\mathcal{D}^\mathrm{op}}\]
>
> The composite \\(\psi\phi\\) of two profunctors \\(\phi\colon \mathcal{C}\nrightarrow \mathcal{D}\\) and \\(\psi\colon \mathcal{D}\nrightarrow E\\)
> is given by
>
> \[ \psi\phi=\mathrm{Lan}_{Y_\mathcal{D}}(\hat{\psi})\circ\hat\phi\]
>
> where \\(\mathrm{Lan}\_{Y\_\mathcal{D}}(\hat{\psi})\\) is the left Kan extension of the functor \\(\hat{\psi}\\) along the (covariant) Yoneda functor \\(Y\_{\mathcal{D}} \colon \mathcal{D}\to\mathrm{Set}^{\mathcal{D}^\mathrm{op}}\\) of \\(\mathcal{D}\\)

I think the Yoneda functor is related to the \\(\mathbf{hom}\\) functor John introduced in [Lecture 52](https://forum.azimuthproject.org/discussion/2273/lecture-52-the-hom-functor).