Section Equalizer.
Context {C : Category} {a b : Obj} (f g : a –≻ b).
(** given two parallel arrows f,g : a -> b, their equalizer is an object e together with an arrow eq : e -> a such that f ∘ eq = g ∘ eq such that for any other object z and eqz : z -> a that we have f ∘ eqz = g ∘ eqz, there is a unique arrow h : z -> e that makes the following diagram commute:  # <pre>  eqz /—————————————————\     f |                 ↓  ———————> z ———–> e ——————> a          b ∃!h      eq       ———–——–> g </pre> # *)
Local Open Scope morphism_scope.
Record Equalizer : Type :=
{
equalizer : C;
equalizer_morph : equalizer –≻ a;
equalizer_morph_com : f ∘ equalizer_morph = g ∘ equalizer_morph;
equalizer_morph_ex (e' : Obj) (eqm : e' –≻ a) :
f ∘ eqm = g ∘ eqm → e' –≻ equalizer;
equalizer_morph_ex_com (e' : Obj) (eqm : e' –≻ a)
(eqmc : f ∘ eqm = g ∘ eqm)
: equalizer_morph ∘ (equalizer_morph_ex e' eqm eqmc) = eqm;
equalizer_morph_unique (e' : Obj) (eqm : e' –≻ a)
(com : f ∘ eqm = g ∘ eqm) (u u' : e' –≻ equalizer)
: equalizer_morph ∘ u = eqm → equalizer_morph ∘ u' = eqm → u = u'
}.
Coercion equalizer : Equalizer >-> Obj.

(** Equalizers are unique up to isomorphism. *)
Theorem Equalizer_iso (e1 e2 : Equalizer) : (e1 ≃ e2)%isomorphism.