Try doing some basic maths questions in the Lean Theorem Prover. Functions, real numbers, equivalence relations and groups. Click on README.md and then on "Open in CoCalc with one click".
License: APACHE
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import category_theory.opposites
import category_theory.products.basic
/-!
The hom functor, sending `(X, Y)` to the type `X ⟶ Y`.
-/
universes v u
open opposite
open category_theory
namespace category_theory.functor
variables (C : Type u) [𝒞 : category.{v} C]
include 𝒞
/-- `functor.hom` is the hom-pairing, sending (X,Y) to X → Y, contravariant in X and covariant in Y. -/
definition hom : Cᵒᵖ × C ⥤ Type v :=
{ obj := λ p, unop p.1 ⟶ p.2,
map := λ X Y f, λ h, f.1.unop ≫ h ≫ f.2 }
@[simp] lemma hom_obj (X : Cᵒᵖ × C) : (hom C).obj X = (unop X.1 ⟶ X.2) := rfl
@[simp] lemma hom_pairing_map {X Y : Cᵒᵖ × C} (f : X ⟶ Y) :
(hom C).map f = λ h, f.1.unop ≫ h ≫ f.2 := rfl
end category_theory.functor