English
Given functors F: A → B and G: C → D, there is a cartesian product functor F × G: A × C → B × D which sends (a,c) to (F(a), G(c)) and acts on morphisms componentwise: (f,g) ↦ (F(f), G(g)).
Русский
Если заданы функторы F: A → B и G: C → D, существует пронзательное произведение функторов F × G: A × C → B × D, отправляющее (a,c) в (F(a), G(c)) и действующее на морфизма по компонентам: (f,g) ↦ (F(f), G(g)).
LaTeX
$$$\\mathrm{prod} (F,G): A \\times C \\to B \\times D$, with $\\mathrm{obj}(a,c)=(F(a),G(c))$ and $\\mathrm{map}(f,g)=(F(f),G(g))$.$$
Lean4
/-- The Cartesian product of two functors. -/
@[simps]
def prod (F : A ⥤ B) (G : C ⥤ D) : A × C ⥤ B × D
where
obj X := (F.obj X.1, G.obj X.2)
map
f :=
(F.map f.1, G.map f.2)
/- Because of limitations in Lean 3's handling of notations, we do not setup a notation `F × G`.
You can use `F.prod G` as a "poor man's infix", or just write `functor.prod F G`. -/
