Profunctor

Remarks

Conventions differ: some authors call the same functor a profunctor from $\mathcal{D}$ to $\mathcal{C}$. For small $\mathcal{D}$ it is the same as a functor $\mathcal{C} \to [\mathcal{D}^{\mathrm{op}}, \mathbf{Set}]$, from $\mathcal{C}$ to presheaves on $\mathcal{D}$. Example: the hom functor $\Hom_{\mathcal{C}} \colon \mathcal{C}^{\mathrm{op}} \times \mathcal{C} \to \mathbf{Set}$ is a profunctor from $\mathcal{C}$ to $\mathcal{C}$, the identity profunctor. Example: a functor $F \colon \mathcal{C} \to \mathcal{D}$ gives the profunctor $(d, c) \mapsto \Hom_{\mathcal{D}}(d, F(c))$ from $\mathcal{C}$ to $\mathcal{D}$. Non-example: for the same $F$, the functor $(d, c) \mapsto \Hom_{\mathcal{D}}(F(c), d)$ is covariant in $d$ and contravariant in $c$. So it is not a profunctor from $\mathcal{C}$ to $\mathcal{D}$; it is a profunctor from $\mathcal{D}$ to $\mathcal{C}$. For $P \colon \mathcal{C} \nrightarrow \mathcal{D}$ and $Q \colon \mathcal{D} \nrightarrow \mathcal{E}$ with $\mathcal{D}$ small, the composite is given by the coend $(Q \circ P)(e, c) = \int^{d \in \mathcal{D}} Q(e, d) \times P(d, c)$. Composition is associative and unital only up to natural isomorphism.