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.
Global Instance BimapId_Coproduct : BimapId C bif.
Global Instance BimapCat_Coproduct : BimapCat C bif.
(** The coproduct is commutative *)
Global Instance SwapInvolutive_Coproduct {a b : obj}
: SemiIso C (swap_ a b) swap.
Global Instance AssocLAssocL_Coproduct : AssocLAssocL C bif.
Context {i : obj}.
Context {Initial_i : Initial C i}.
Context {InitialObject_i : InitialObject C i}.
(* Vanishing. These two loops:  [[ +----------+ | +-----+  | | | ### |  | | +-###-+I | +---###----+J A-----###-------B ### ]]  ... can be rewired as a single one:   [[ +-------+ |  ###  | +--###--+(I+J) +--###--+ A-----###-----B ### ]]  *)
(* Here we show that we can implement [ITree.cat] using [bimap], [loop], and composition with the monoidal natural isomorphisms. *)

(* [cat_from_loop]:  +-------------+ |             | +---\/---ab---+ -------/\---bc-------  is equivalent to  ----ab---bc---- *)
Theorem cat_from_loop {a b c} (ab : C a b) (bc : C b c)
: loop (swap >>> bimap ab bc) ⩯ ab >>> bc.