Definition cat_eq_2 (C D : precategory_data) : UU := ∑ (F : ∑ (Fo : ob C = ob D), data_cat_eq_2 C D Fo), (∏ (a : C), eqweqmap (pr2 F a a) (identity a) = identity (eqweqmap (pr1 F) a)) × (∏ (a b c : C) (f : C⟦a,b⟧) (g : C⟦b,c⟧), eqweqmap (pr2 F a c) (f · g) = eqweqmap (pr2 F a b) f · eqweqmap (pr2 F b c) g).
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.catiso.