proposition: Lemma homset_conj_inv_natural_precomp {c : C} {c' : C'} {g : C⟦c, G c'⟧} {c'' : C} {f : C⟦c'', c⟧} {d : D c} {d' : D' c'} {d'' : D c''} (gg : d -->[g] GG _ d') (ff : d'' -->[f] d) : homset_conj_inv _ _ _ (ff ;; gg) = transportb _ (φ_adj_inv_natural_precomp A _ _ g _ f) (♯ FF ff ;; homset_conj_inv _ _ _ gg).
proof: Proof. unfold homset_conj_inv. rewrite disp_functor_comp. rewrite assoc_disp. unfold transportb. rewrite mor_disp_transportf_postwhisker. rewrite transport_f_f. apply maponpaths_2, homset_property. Defined.
imports: UniMath.Foundations.Sets UniMath.MoreFoundations.PartA UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Equivalences