Lemma is_functor_functor_fix_snd_arg_data: is_functor functor_fix_snd_arg_data. Proof. split. + intros c. unfold functor_fix_snd_arg_data; simpl. unfold functor_fix_snd_arg_mor; simpl. unfold functor_fix_snd_arg_ob; simpl. assert (functor_id_inst := functor_id F). rewrite <- functor_id_inst. apply maponpaths. apply idpath. + intros c c' c'' f g. unfold functor_fix_snd_arg_data; simpl. unfold functor_fix_snd_arg_mor; simpl. assert (functor_comp_inst := @functor_comp _ _ F (make_dirprod c d) (make_dirprod c' d) (make_dirprod c'' d)). rewrite <- functor_comp_inst. apply maponpaths. unfold compose at 2. unfold precategory_binproduct; simpl. rewrite id_left. apply idpath. Qed.
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.FunctorCategory. Require Import UniMath.CategoryTheory.opp_precat.