Section CoproductFacts.
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 {Category_C : Category C}.
Context {bif : binop obj}.
Context {Case_C : Case C bif}
{Inl_C : Inl C bif}
{Inr_C : Inr C bif}.
Context {Coproduct_C : Coproduct C bif}.

(** We can prove the equivalence of morphisms on coproducts by case analysis. *)
Lemma coprod_split {a b c} (f g : C (bif a b) c) :
(inl_ >>> f ⩯ inl_ >>> g) ->
(inr_ >>> f ⩯ inr_ >>> g) ->
f ⩯ g.