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Require Export Corelib.Classes.Morphisms.
Require Export Corelib.Relations.Relation_Definitions.
Require Export Corelib.Setoids.Setoid.
Set Universe Polymorphism.
Reserved Notation "a ~> b" (at level 90, right associativity).
Class Category := {
ob : Type;
uhom := Type : Type;
hom : ob -> ob -> uhom where "a ~> b" := (hom a b);
compose : forall {A B C}, (B ~> C) -> (A ~> B) -> (A ~> C);
equiv : forall {A B}, relation (A ~> B);
is_equiv : forall {A B}, @Equivalence (A ~> B) equiv;
comp_respects : forall {A B C},
Proper (@equiv B C ==> @equiv A B ==> @equiv A C) (@compose A B C);
}.
Add Parametric Morphism `{Category} {A B C} : (@compose _ A B C) with
signature equiv ==> equiv ==> equiv as compose_mor.
Proof. apply comp_respects. Qed.
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