proposition: Lemma disp_functor_comp_two_disp {x₁ x₂ x₃ y₁ y₂ y₃ : C} (xx := (x₁ ,, y₁) : category_binproduct C C) (yy := (x₂ ,, y₂) : category_binproduct C C) (zz := (x₃ ,, y₃) : category_binproduct C C) {f₁ : x₁ --> x₂} {f₂ : x₂ --> x₃} {g₁ : y₁ --> y₂} {g₂ : y₂ --> y₃} {xy₁ : D₁ x₁ y₁} {xy₂ : D₁ x₂ y₂} {xy₃ : D₁ x₃ y₃} (fg₁ : xy₁ -->[ f₁ ][ g₁ ] xy₂) (fg₁' := fg₁ : @mor_disp _ (twosided_disp_cat_to_disp_cat C C D₁) xx yy xy₁ xy₂ (f₁ ,, g₁)) (fg₂ : xy₂ -->[ f₂ ][ g₂ ] xy₃) (fg₂' := fg₂ : @mor_disp _ (twosided_disp_cat_to_disp_cat C C D₁) yy zz xy₂ xy₃ (f₂ ,, g₂)) (h := fg₁ ;;2 fg₂ : @mor_disp _ (twosided_disp_cat_to_disp_cat C C D₁) xx zz xy₁ xy₃ (f₁ · f₂ ,, g₁ · g₂)) : (♯ F h = ♯F fg₁' ;;2 ♯F fg₂')%mor_disp.
proof: Proof. exact (disp_functor_comp F fg₁' fg₂'). Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Core.Setcategories UniMath.CategoryTheory.Categories.CategoryOfSetCategories UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.DisplayedCatEq UniMath.CategoryTheory.DisplayedCats.Examples.DispFunctorPair UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness UniMath.CategoryTheory.TwoSidedDisplayedCats.TransportLaws