@include logic.pg

section Category
    variable
        (Obj  : *)
        (Hom  : Obj -> Obj -> *)
        (id   : forall (A     : Obj), Hom A A)
        (comp : forall (A B C : Obj), Hom A B -> Hom B C -> Hom A C);

    hypothesis
        (id_l  : forall (A B     : Obj) (f : Hom A B), eq (Hom A B) (comp A A B (id A) f) f)
        (id_r  : forall (A B     : Obj) (f : Hom B A), eq (Hom A B) (comp B A A f (id A)) f)
        (assoc : forall (A B C D : Obj) (f : Hom A B) (g : Hom B C) (h : Hom C D),
            eq (Hom A D)
                (comp A B D f (comp B C D g h))
                (comp A C D (comp A B C f g) h));

    def initial  (A : Obj) := forall (B : Obj), exists_uniq (Hom A B) (fun (f : Hom A B) => true);
    def terminal (A : Obj) := forall (B : Obj), exists_uniq (Hom B A) (fun (f : Hom B A) => true);

    section Inverses
        variable
            (A B : Obj)
            (f : Hom A B)
            (g : Hom B A);

        def inv_l := eq (Hom A A) (comp A B A f g) (id A);
        def inv_r := eq (Hom B B) (comp B A B g f) (id B);

        def inv := and inv_l inv_r;
    end Inverses

    def isomorphic (A B : Obj) :=
        exists (Hom A B) (fun (f : Hom A B) =>
            exists (Hom B A) (fun (g : Hom B A) => 
                inv A B f g));
end Category