Qed.
Class PreMonoidalFunctor
-`(PM1:PreMonoidalCat(C:=C1)(I:=I1))
-`(PM2:PreMonoidalCat(C:=C2)(I:=I2))
- (fobj : C1 -> C2 ) :=
-{ mf_F :> Functor C1 C2 fobj
+`(PM1 : PreMonoidalCat(C:=C1)(I:=I1))
+`(PM2 : PreMonoidalCat(C:=C2)(I:=I2))
+ {fobj : C1 -> C2 }
+ (F : Functor C1 C2 fobj ) :=
+{ mf_F := F
; mf_i : I2 ≅ mf_F I1
; mf_first : ∀ a, mf_F >>>> bin_first (mf_F a) <~~~> bin_first a >>>> mf_F
; mf_second : ∀ a, mf_F >>>> bin_second (mf_F a) <~~~> bin_second a >>>> mf_F
`{PM1 :PreMonoidalCat(C:=C1)(I:=I1)}
`{PM2 :PreMonoidalCat(C:=C2)(I:=I2)}
{fobj12:C1 -> C2 }
- (PMF12 :PreMonoidalFunctor PM1 PM2 fobj12)
+ {PMFF12:Functor C1 C2 fobj12 }
+ (PMF12 :PreMonoidalFunctor PM1 PM2 PMFF12)
`{PM3 :PreMonoidalCat(C:=C3)(I:=I3)}
{fobj23:C2 -> C3 }
- (PMF23 :PreMonoidalFunctor PM2 PM3 fobj23).
+ {PMFF23:Functor C2 C3 fobj23 }
+ (PMF23 :PreMonoidalFunctor PM2 PM3 PMFF23).
Definition compose_mf := PMF12 >>>> PMF23.
Implicit Arguments id [[Ob][Hom][Category]].
(* this proof is really gross; I will write a better one some other day *)
- Instance PreMonoidalFunctorsCompose : PreMonoidalFunctor PM1 PM3 (fobj23 ○ fobj12) :=
+ Instance PreMonoidalFunctorsCompose : PreMonoidalFunctor PM1 PM3 compose_mf :=
{ mf_i := compose_mf_i
- ; mf_F := compose_mf
; mf_first := compose_mf_first
; mf_second := compose_mf_second }.