-{ br_swap : forall a b, a⊗b ≅ b⊗a
-; triangleb : forall a:C, #(pmon_cancelr mc a) ~~ #(br_swap a (pmon_I(PreMonoidalCat:=mc))) >>> #(pmon_cancell mc a)
-; hexagon1 : forall {a b c}, #(pmon_assoc mc _ _ _) >>> #(br_swap a _) >>> #(pmon_assoc mc _ _ _)
- ~~ #(br_swap _ _) ⋉ c >>> #(pmon_assoc mc _ _ _) >>> b ⋊ #(br_swap _ _)
-; hexagon2 : forall {a b c}, #(pmon_assoc mc _ _ _)⁻¹ >>> #(br_swap _ c) >>> #(pmon_assoc mc _ _ _)⁻¹
- ~~ a ⋊ #(br_swap _ _) >>> #(pmon_assoc mc _ _ _)⁻¹ >>> #(br_swap _ _) ⋉ b
+{ br_niso : forall a, bin_first a <~~~> bin_second a
+; br_swap := fun a b => ni_iso (br_niso b) a
+; triangleb : forall a:C, #(pmon_cancelr mc a) ~~ #(br_swap a (pmon_I(PreMonoidalCat:=mc))) >>> #(pmon_cancell mc a)
+; hexagon1 : forall {a b c}, #(pmon_assoc mc _ _ _) >>> #(br_swap a _) >>> #(pmon_assoc mc _ _ _)
+ ~~ #(br_swap _ _) ⋉ c >>> #(pmon_assoc mc _ _ _) >>> b ⋊ #(br_swap _ _)
+; hexagon2 : forall {a b c}, #(pmon_assoc mc _ _ _)⁻¹ >>> #(br_swap _ c) >>> #(pmon_assoc mc _ _ _)⁻¹
+ ~~ a ⋊ #(br_swap _ _) >>> #(pmon_assoc mc _ _ _)⁻¹ >>> #(br_swap _ _) ⋉ b