> 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).