Definition family_functor_data (I : UU) {A B : I -> category} (F : ∏ (i : I), (A i) ⟶ (B i)) : functor_data (product_category A) (product_category B). Proof. use tpair. - intros a i; apply (F i (a i)). - intros a b f i; apply (# (F i) (f i)). Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.opp_precat.