Qed.
Definition roundtrip_lemma
- (K:SurjectiveEnrichment) (C:MonicMonoidalEnrichment) (reification : Reification K C (me_i C))
+ `(K:SurjectiveEnrichment se) `(C:MonicMonoidalEnrichment e ce) (reification : Reification K C (me_i C))
:= roundtrip_lemma' (RepresentableFunctor C (me_i C)) (ffme_mf_full C) (ffme_mf_faithful C) (step1_functor K C reification).
Lemma roundtrip_reification_to_reification
- (K:SurjectiveEnrichment) (C:MonicMonoidalEnrichment) (reification : Reification K C (me_i C))
+ `(K:SurjectiveEnrichment se) `(C:MonicMonoidalEnrichment e ce) (reification : Reification K C (me_i C))
: reification ≃ reification_from_garrow K C (garrow_from_reification K C reification).
simpl.
unfold mon_f.
unfold garrow_functor.
apply (if_comp(F2:=(step1_functor K C reification >>>> InclusionFunctor _ (FullImage (RepresentableFunctor C (me_i C)))))).
- apply step1_niso.
+ apply (@step1_niso _ K _ _ C reification).
apply (if_inv (roundtrip_lemma K C reification)).
Qed.
- (* FIXME: show that the R-functors are naturally isomorphic as well; should follow pretty easily from the proof for Rstar *)
Lemma roundtrip_garrow_to_garrow
- (K:SurjectiveEnrichment) (C:MonicMonoidalEnrichment) (garrow : GeneralizedArrow K C)
+ `(K:SurjectiveEnrichment se) `(C:MonicMonoidalEnrichment e ce) (garrow : GeneralizedArrow K C)
: garrow ≃ garrow_from_reification K C (reification_from_garrow K C garrow).
apply (ffc_functor_weakly_monic _ (ffme_mf_conservative C)).
apply if_inv.
unfold garrow_from_reification.
unfold garrow_functor.
unfold step2_functor.
- apply roundtrip_lemma.
+ set (@roundtrip_lemma _ K _ _ C) as q.
+ apply (q (reification_from_garrow K C garrow)).
apply if_inv.
apply (step1_niso K C (reification_from_garrow K C garrow)).
Qed.