Lemma. Promonads are identity on objects functors [mde-0009]

Let \((T, \eta , \mu )\) be a promonad on a category \(ℂ\), then there is an identity-on-objects functor \(ℂ \to \textsf {kl}(T)\) whose action on hom-sets is given by the components \(\eta _{A,B} : ℂ(A;B) \to \textsf {kl}(T)(A;B)\), where \(\textsf {kl}(T)\) is the Kleisli category of the promonad. This is functorial by the definition of \(\textsf {kl}(T)\).

Conversely, let \(F : ℂ \to \mathbb {D}\) be an identity-on-objects functor. Then \(F^{*}F_{*} : ℂ → ℂ\) is a endoprofunctor, where \(F_{*} := \lambda d,c . \mathbb {D}(d; Fc)\) and \(F^{*} := \lambda d,c . \mathbb {D}(Fc; d)\) are the (co)representable profunctors determined by \(F\). The components of the unit are given by the action of \(F\) on hom-sets, and multiplication is given by composition in \(\mathbb {D}\).