Proposition fully_faithful_enriched_factorization_precomp {C₁ C₂ C₃ : category} {F : C₁ ⟶ C₃} {G : C₁ ⟶ C₂} {H : C₂ ⟶ C₃} {α : nat_z_iso F (G ∙ H)} {V : monoidal_cat} {E₁ : enrichment C₁ V} {E₂ : enrichment C₂ V} {E₃ : enrichment C₃ V} (EF : functor_enrichment F E₁ E₃) (EG : functor_enrichment G E₁ E₂) (EH : functor_enrichment H E₂ E₃) (Hα : nat_trans_enrichment α EF (functor_comp_enrichment EG EH)) (HEF : fully_faithful_enriched_functor EF) (HEH : fully_faithful_enriched_functor EH) : fully_faithful_enriched_functor EG. Proof. exact (fully_faithful_enriched_precomp _ _ (fully_faithful_enriched_nat_z_iso Hα HEF) HEH). Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.EnrichedCats.Enrichment. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentFunctor. Require Import UniMath.CategoryTheory.EnrichedCats.EnrichmentTransformation. Require Import UniMath.CategoryTheory.Monoidal.Categories.