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}.

(** Commute [cat] and [case_]. *)
Lemma cat_case
{a b c d} (ac : C a c) (bc : C b c) (cd : C c d)
: case_ ac bc >>> cd ⩯ case_ (ac >>> cd) (bc >>> cd).