proposition: Lemma lift_functor_along_comm_prod (H : functor (category_binproduct C C) C) : nat_z_iso (lift_functor_along D FFF FFF_eso FFF_ff ((pair_functor H (functor_identity C) ∙ H) ∙ F1)) (pair_functor (lift_functor_along D FF FF_eso FF_ff (functor_composite H F1) ) (functor_identity D) ∙ (lift_functor_along D FF FF_eso FF_ff (functor_composite H F1)) ) .
proof: Proof. use nat_z_iso_comp. 2: { exact (lift_functor_along_comp DD D F1 FFF_eso FFF_ff FF_eso FF_ff (pair_functor H (functor_identity C)) H). } apply post_whisker_nat_z_iso. use nat_z_iso_comp. 2 : { apply lift_functor_along_pair. } use nat_z_iso_between_pair. - use make_nat_z_iso. 2: apply is_nat_z_iso_nat_trans_id. - apply lift_functor_along_id. Defined.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Equivalences.FullyFaithful UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.precomp_ess_surj UniMath.CategoryTheory.precomp_fully_faithful UniMath.CategoryTheory.PrecategoryBinProduct