English
mkNatIso and the preimageIso under a fully faithful functor agree up to the appropriate isomorphism; this identifies two canonical ways of obtaining isomorphisms from CatCospanTransform data.
Русский
mkNatIso и предобразная изоморфия под полностью-фиделитным функтором согласуются, давая идентификацию двух канонических способов получения изоморфий из данных CatCospanTransform.
LaTeX
$$$ mkNatIso e_1 e_2 coh = (functorEquiv F G X).fullyFaithfulFunctor.preimageIso (CategoricalPullback.mkIso e_1 e_2 (by simpa [functorEquiv, toCatCommSqOver] using coh)) $$$
Lean4
theorem transform_map_associator {A₃ : Type u₁₀} {B₃ : Type u₁₁} {C₃ : Type u₁₂} [Category.{v₁₀} A₃] [Category.{v₁₁} B₃]
[Category.{v₁₂} C₃] {F₃ : A₃ ⥤ B₃} {G₃ : C₃ ⥤ B₃} (X : Type u₁₃) [Category.{v₁₃} X]
(ψ : CatCospanTransform F G F₁ G₁) (φ : CatCospanTransform F₁ G₁ F₂ G₂) (τ : CatCospanTransform F₂ G₂ F₃ G₃) :
(transform X).map (α_ ψ φ τ).hom =
(transformObjComp X (ψ.comp φ) τ).hom ≫
whiskerRight (transformObjComp X ψ φ).hom (transform X |>.obj τ) ≫
((transform X |>.obj ψ).associator (transform X |>.obj φ) (transform X |>.obj τ)).hom ≫
whiskerLeft (transform X |>.obj ψ) (transformObjComp X φ τ).inv ≫ (transformObjComp X ψ (φ.comp τ)).inv :=
by cat_disch
