-- Module: Mathlib.GroupTheory.FreeGroup.NielsenSchreier
Goal:
G A : Type u
inst : Group G
inst_1 : IsFreeGroup G
inst_2 : MulAction G A
X : Type u
inst_3 : Group X
f : Labelling (Generators (ActionCategory G A)) X
f' : IsFreeGroup.Generators G → (A → X) ⋊[mulAutArrow] G :=
  fun (e : IsFreeGroup.Generators G) ↦
    ⟨fun (b : A) ↦
      f (a :=
        Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) ()
          ((IsFreeGroup.of e)⁻¹ • b))
        (b := Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) () b)
        (Subtype.mk (p := fun (e_1 : IsFreeGroup.Generators G) ↦
          HSMul.hSMul (β := A) (IsFreeGroup.of e_1)
              (ActionCategory.back (M := G)
                (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) ()
                  ((IsFreeGroup.of e)⁻¹ • b))) =
            ActionCategory.back (M := G)
              (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) () b))
          e ⋯),
      IsFreeGroup.of e⟩
F' : G →* (A → X) ⋊[mulAutArrow] G
hF' : ∀ (a : IsFreeGroup.Generators G), (F' : (a : G) → (A → X) ⋊[mulAutArrow] G) (IsFreeGroup.of a) = f' a
uF' :
  ∀ (y : G →* (A → X) ⋊[mulAutArrow] G),
    (fun (F : G →* (A → X) ⋊[mulAutArrow] G) ↦
          ∀ (a : IsFreeGroup.Generators G), (F : (a : G) → (A → X) ⋊[mulAutArrow] G) (IsFreeGroup.of a) = f' a)
        y →
      y = F'
a : A
b : Functor.obj (actionAsFunctor G A) PUnit.unit
e : IsFreeGroup.Generators G
h : IsFreeGroup.of e • a = b
refine_2 :
  SemidirectProduct.left (G := G) (φ := mulAutArrow)
      ((F' : (a : G) → (A → X) ⋊[mulAutArrow] G)
        (IsFreeGroup.of
          (↑(Subtype.mk (p := fun (e : IsFreeGroup.Generators G) ↦
                HSMul.hSMul (β := A) (IsFreeGroup.of e)
                    (ActionCategory.back (M := G)
                      (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c)
                        PUnit.unit a)) =
                  ActionCategory.back
                    (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c)
                      PUnit.unit b))
                e h) :
            IsFreeGroup.Generators G)))
      (ActionCategory.back
        (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit b)) =
    f (a := Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit a)
      (b := Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit b)
      (Subtype.mk (p := fun (e : IsFreeGroup.Generators G) ↦
        HSMul.hSMul (β := A) (IsFreeGroup.of e)
            (ActionCategory.back (M := G)
              (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit
                a)) =
          ActionCategory.back
            (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit b))
        e h)
⊢ (uncurry F' ⋯).map
      (Subtype.mk (p :=
        fun
          (f :
            Sigma.fst (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c)
                (have this : ActionCategory G A :=
                  Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit
                    a;
                this) ⟶
              (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit
                    b).fst
                (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c)) ↦
        (ConcreteCategory.hom (FC := TypeCat.Fun) (CategoryTheory.Functor.map (actionAsFunctor G A) f) :
              (a :
                  (fun (X : Type u) ↦ X)
                    ((actionAsFunctor G A).obj
                      (have this := ⟨PUnit.unit, a⟩;
                        this).fst)) →
                (fun (X : Type u) ↦ X) ((actionAsFunctor G A).obj ⟨PUnit.unit, b⟩.fst))
            (Sigma.snd (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c)
              (have this : ActionCategory G A :=
                Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit a;
              this)) =
          (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit b).snd
            (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c))
        (IsFreeGroup.of
          (↑(Subtype.mk (p := fun (e : IsFreeGroup.Generators G) ↦
                HSMul.hSMul (β := A) (IsFreeGroup.of e)
                    (ActionCategory.back (M := G)
                      (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c)
                        PUnit.unit a)) =
                  ActionCategory.back
                    (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c)
                      PUnit.unit b))
                e h) :
            IsFreeGroup.Generators G))
        ⋯) =
    f (a := Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit a)
      (b := Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit b)
      (Subtype.mk (p := fun (e : IsFreeGroup.Generators G) ↦
        HSMul.hSMul (β := A) (IsFreeGroup.of e)
            (ActionCategory.back (M := G)
              (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit
                a)) =
          ActionCategory.back
            (Sigma.mk (β := fun (c : CategoryTheory.SingleObj G) ↦ Functor.obj (actionAsFunctor G A) c) PUnit.unit b))
        e h)

Tactic:
change (F' (IsFreeGroup.of _)).left _ = _

Premises used: IsFreeGroup.of, SemidirectProduct.left, DFunLike.coe