Definition functor_total_category_of_set_functor_on_comp {C₁ C₂ C₃ : setcategory} {F₁ : C₁ ⟶ C₂} {F₂ : C₂ ⟶ C₃} {G₁ : C₁ ⟶ cat_of_setcategory} {G₂ : C₂ ⟶ cat_of_setcategory} {G₃ : C₃ ⟶ cat_of_setcategory} (α : G₁ ⟹ F₁ ∙ G₂) (β : G₂ ⟹ F₂ ∙ G₃) : functor_total_category_of_set_functor F₁ α ∙ functor_total_category_of_set_functor F₂ β ⟹ functor_total_category_of_set_functor (F₁ ∙ F₂) (nat_trans_comp _ _ _ α (pre_whisker F₁ β)). Proof. use make_nat_trans. - exact (functor_total_category_of_set_functor_on_comp_data α β). - exact (functor_total_category_of_set_functor_on_comp_is_nat_trans α β). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Categories.CategoryOfSetCategories. Require Import UniMath.CategoryTheory.GrothendieckConstruction.TotalCategory. Section FunctorTotalCategoryFromSetFunctor. End FunctorTotalCategoryFromSetFunctor. Section NatTransTotalCategoryFromNatTrans. End NatTransTotalCategoryFromNatTrans.