feat(hott) formalize book lemma 9.9.1: essentially surjective functors induce faithful functors in the functor category

hott/algebra/category/constructions/functor.hlean

@@ -6,7 +6,7 @@ Authors: Floris van Doorn, Jakob von Raumer
 Functor precategory and category
 -/
 
-import ..nat_trans ..category .opposite
+import .opposite ..functor.attributes
 
 open eq category is_trunc nat_trans iso is_equiv category.hom
 
@@ -372,6 +372,15 @@ namespace functor
     { intro G H I η θ, reflexivity},
   end
 
+  definition faithful_precomposition_of_essentially_surjective [instance] 
+    {C D E} {H : C ⇒ D} [HH : essentially_surjective H] : faithful (precomposition_functor E H) :=
+  begin
+    intro F G γ δ Hγδ, apply nat_trans_eq, intro b,
+    induction HH b with Hb, induction Hb with a f,
+    refine naturality_iso_right γ f ⬝ _ ⬝ (naturality_iso_right δ f)⁻¹,
+    apply ap (λ x, _ ∘ natural_map x a ∘ _) Hγδ,
+  end
+
   definition postcomposition_functor [constructor] {C D} (E) (F : C ⇒ D)
     : C ^c E ⇒ D ^c E :=
   begin

hott/algebra/category/functor/attributes.hlean

@@ -8,7 +8,7 @@ Attributes of functors (full, faithful, split essentially surjective, ...)
 Adjoint functors, isomorphisms and equivalences have their own file
 -/
 
-import ..constructions.functor function arity
+import .basic function arity
 
 open eq functor trunc prod is_equiv iso equiv function is_trunc
 

hott/algebra/category/nat_trans.hlean

@@ -47,6 +47,13 @@ namespace nat_trans
     (g : d ⟶ d') : constant_functor C d ⟹ constant_functor C d' :=
   mk (λc, g) (λc c' f, !id_comp_eq_comp_id)
 
+  open iso
+  definition naturality_iso_left (η : F ⟹ G) {a b : C} (f : a ≅ b) : η a = (G f)⁻¹ ∘ η b ∘ F f :=
+  by apply eq_inverse_comp_of_comp_eq; apply naturality
+
+  definition naturality_iso_right (η : F ⟹ G) {a b : C} (f : a ≅ b) : η b = G f ∘ η a ∘ (F f)⁻¹ :=
+  by refine _⁻¹ ⬝ !assoc⁻¹; apply comp_inverse_eq_of_eq_comp; apply naturality
+
   definition nat_trans_mk_eq {η₁ η₂ : Π (a : C), hom (F a) (G a)}
     (nat₁ : Π (a b : C) (f : hom a b), G f ∘ η₁ a = η₁ b ∘ F f)
     (nat₂ : Π (a b : C) (f : hom a b), G f ∘ η₂ a = η₂ b ∘ F f)