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Callan McGill (Apr 09 2022 at 03:30):There is a variant on the usual yoneda lemma where we do it relative to a category: say we have a functor F : C -> D and c : C, d : D then we get a natural bijection between morphisms F c -> d and Nat[Hom(-, c), F ↓ d). One can recover the usual Yoneda lemma as a special case by writing a presheaf P : C^op -> Set as P~ : C -> Set^op and then taking D = Set^op and d = * in which case we get that P~ ↓ * is the same as P~. This is also very related to the usual set-up for the limit/colimit formula for pointwise kan extensions. I am wondering if there are some more general povs from which to understand this sort of relative construction?
Christian Williams (Apr 09 2022 at 18:23):for any profunctor P:C->D, there is an isomorphism Nat[C(-,c), P(-,d)] = P(c,d). this is a general fact that holds in proarrow equipments, see for example Proarrows I by Wood, Prop 3 (letting "b" be a constant functor at c).