Composition of Profunctors

Let $P:\mathcal{D}^{\mathrm{op}}\times\mathcal{E}\to\mathrm{Sets}$, $Q:\mathcal{C}^{\mathrm{op}}\times\mathcal{D}\to\mathrm{Sets}$ be a pair of Profunctors. Their composite $P\otimes Q:\mathcal{C}^{\mathrm{op}}\times\mathcal{E}\to\mathrm{Sets}$ is a profunctor with

• $(P\otimes Q)(c,e)=\Sigma(d:\mathcal{D}).\;P(c,d)\times Q(d,e)$, quotiented by the relation that $(d,p,q)\sim(d^{\prime},p^{\prime},q^{\prime})$ when there exists a $f:\mathcal{D}(d,d^{\prime})$ with $p(f,\mathrm{id})=p^{\prime}$ and $q=q^{\prime}(\mathrm{id},f)$.
• Actions are given by actions on $P$ and $Q$.

Alternatively, profunctor composition can be written as the Coend

If we view profunctors as Bimodules of categories, then the composition of profunctors is the Tensor Product of Bimodules.