Definition Hom_comp_assoc :=
  forall (a b c d : Ob) (f : Hom a b) (g : Hom b c) (h : Hom c d),
  Equal (Hom_comp a b d (couple (Hom_comp b c d (couple h g)) f))
    (Hom_comp a c d (couple h (Hom_comp a b c (couple g f)))).Definition Hom_comp_unit_l :=
  forall (a b : Ob) (f : Hom a b),
  Equal (Hom_comp a b b (couple (Hom_id b) f)) f.Definition Hom_comp_unit_r :=
  forall (a b : Ob) (f : Hom a b),
  Equal (Hom_comp a a b (couple f (Hom_id a))) f.Record category : Type := 
  {Ob :> Type;
   Hom : Ob -> Ob -> Setoid;
   Hom_comp : forall a b c : Ob, MAP (cart (Hom b c) (Hom a b)) (Hom a c);
   Hom_id : forall a : Ob, Hom a a;
   Hom_comp_assoc_prf : Hom_comp_assoc Hom_comp;
   Hom_comp_unit_l_prf : Hom_comp_unit_l Hom_comp Hom_id;
   Hom_comp_unit_r_prf : Hom_comp_unit_r Hom_comp Hom_id}.Record category : Type := 
  {Ob :> Type;
   Hom : Ob -> Ob -> Setoid;
   Hom_comp : forall a b c : Ob, MAP (cart (Hom b c) (Hom a b)) (Hom a c);
   Hom_id : forall a : Ob, Hom a a;
   Hom_comp_assoc_prf : Hom_comp_assoc Hom_comp;
   Hom_comp_unit_l_prf : Hom_comp_unit_l Hom_comp Hom_id;
   Hom_comp_unit_r_prf : Hom_comp_unit_r Hom_comp Hom_id}.Record category : Type := 
  {Ob :> Type;
   Hom : Ob -> Ob -> Setoid;
   Hom_comp : forall a b c : Ob, MAP (cart (Hom b c) (Hom a b)) (Hom a c);
   Hom_id : forall a : Ob, Hom a a;
   Hom_comp_assoc_prf : Hom_comp_assoc Hom_comp;
   Hom_comp_unit_l_prf : Hom_comp_unit_l Hom_comp Hom_id;
   Hom_comp_unit_r_prf : Hom_comp_unit_r Hom_comp Hom_id}.Record category : Type := 
  {Ob :> Type;
   Hom : Ob -> Ob -> Setoid;
   Hom_comp : forall a b c : Ob, MAP (cart (Hom b c) (Hom a b)) (Hom a c);
   Hom_id : forall a : Ob, Hom a a;
   Hom_comp_assoc_prf : Hom_comp_assoc Hom_comp;
   Hom_comp_unit_l_prf : Hom_comp_unit_l Hom_comp Hom_id;
   Hom_comp_unit_r_prf : Hom_comp_unit_r Hom_comp Hom_id}.Record category : Type := 
  {Ob :> Type;
   Hom : Ob -> Ob -> Setoid;
   Hom_comp : forall a b c : Ob, MAP (cart (Hom b c) (Hom a b)) (Hom a c);
   Hom_id : forall a : Ob, Hom a a;
   Hom_comp_assoc_prf : Hom_comp_assoc Hom_comp;
   Hom_comp_unit_l_prf : Hom_comp_unit_l Hom_comp Hom_id;
   Hom_comp_unit_r_prf : Hom_comp_unit_r Hom_comp Hom_id}.Record category : Type := 
  {Ob :> Type;
   Hom : Ob -> Ob -> Setoid;
   Hom_comp : forall a b c : Ob, MAP (cart (Hom b c) (Hom a b)) (Hom a c);
   Hom_id : forall a : Ob, Hom a a;
   Hom_comp_assoc_prf : Hom_comp_assoc Hom_comp;
   Hom_comp_unit_l_prf : Hom_comp_unit_l Hom_comp Hom_id;
   Hom_comp_unit_r_prf : Hom_comp_unit_r Hom_comp Hom_id}.Record category : Type := 
  {Ob :> Type;
   Hom : Ob -> Ob -> Setoid;
   Hom_comp : forall a b c : Ob, MAP (cart (Hom b c) (Hom a b)) (Hom a c);
   Hom_id : forall a : Ob, Hom a a;
   Hom_comp_assoc_prf : Hom_comp_assoc Hom_comp;
   Hom_comp_unit_l_prf : Hom_comp_unit_l Hom_comp Hom_id;
   Hom_comp_unit_r_prf : Hom_comp_unit_r Hom_comp Hom_id}.Record category : Type := 
  {Ob :> Type;
   Hom : Ob -> Ob -> Setoid;
   Hom_comp : forall a b c : Ob, MAP (cart (Hom b c) (Hom a b)) (Hom a c);
   Hom_id : forall a : Ob, Hom a a;
   Hom_comp_assoc_prf : Hom_comp_assoc Hom_comp;
   Hom_comp_unit_l_prf : Hom_comp_unit_l Hom_comp Hom_id;
   Hom_comp_unit_r_prf : Hom_comp_unit_r Hom_comp Hom_id}.Definition comp_hom (a b c : C) (g : Hom b c) (f : Hom a b) :=
  Hom_comp a b c (couple g f).

(* Specifications: *)
Lemma comp_hom_compatible : forall (a b c : C) (x x' : Hom b c) (y y' : Hom a b), Equal x x' -> Equal y y' -> Equal (comp_hom x y) (comp_hom x' y').
Lemma comp_hom_assoc : forall (a b c d : C) (f : Hom a b) (g : Hom b c) (h : Hom c d), Equal (comp_hom (comp_hom h g) f) (comp_hom h (comp_hom g f)).
Lemma comp_hom_unit_l : forall (a b : C) (f : Hom a b), Equal (comp_hom (Hom_id b) f) f.
Lemma comp_hom_unit_r : forall (a b : C) (f : Hom a b), Equal (comp_hom f (Hom_id a)) f.