Section CategoryFacts.
Context {obj : Type} {C : Hom obj}.
Context {Eq2_C : Eq2 C}.
Context {E_Eq2_C : forall a b, @Equivalence (C a b) eq2}.
Context {Id_C : Id_ C} {Cat_C : Cat C}.
Context {i : obj}.
Context {Initial_i : Initial C i}.
Context {InitialObject_i : InitialObject C i}.

(** The initial morphism is unique. *)
Lemma initial_unique :
forall a (f g : C i a), f ⩯ g.