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}.
Ltac cat_auto :=
unfold_coproduct;
repeat progress (reflexivity || (try cat_auto_simpl); try apply coprod_split).
Ltac cat_auto' :=
repeat intro;
repeat match goal with
| [ |- eq2 _ _ ] => fail 1
| _ => red
end;
cat_auto.

Lemma swap_case (a b c: obj) (f: C a c) (g: C b c)
: swap >>> case_ f g ⩯ case_ g f.