-- Module: Mathlib.GroupTheory.FreeGroup.NielsenSchreier
Goal:
G : Type u
inst : Groupoid G
inst_1 : IsFreeGroupoid G
T : WideSubquiver (Symmetrify (Generators G))
inst_2 : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)
X : Type u
inst_3 : Group X
f :
  (↑((wideSubquiverEquivSetTotal : (a : WideSubquiver (Generators G)) → Set (Quiver.Total (Generators G)))
            (wideSubquiverSymmetrify T))ᶜ :
      Type u) →
    X
f' : Labelling (Generators G) X :=
  fun (a b : Generators G) (e : a ⟶ b) ↦
    if h : e ∈ wideSubquiverSymmetrify T a b then 1
    else
      f
        (Subtype.mk (p := fun (x : Quiver.Total (Generators G)) ↦
          Membership.mem (γ := Set (Quiver.Total (Generators G)))
            (((wideSubquiverEquivSetTotal : (a : WideSubquiver (Generators G)) → Set (Quiver.Total (Generators G)))
                  (wideSubquiverSymmetrify T))ᶜ :
              Set (Quiver.Total (Generators G)))
            x)
          { left := a, right := b, hom := e } h)
F' : G ⥤ CategoryTheory.SingleObj X
hF' : ∀ (a b : Generators G) (g : a ⟶ b), F'.map (of g) = f' g
uF' :
  ∀ (y : G ⥤ CategoryTheory.SingleObj X),
    (fun (F : G ⥤ CategoryTheory.SingleObj X) ↦ ∀ (a b : Generators G) (g : a ⟶ b), F.map (of g) = f' g) y → y = F'
refine_1 :
  (fun (F : End (root' T) →* X) ↦
      ∀
        (a :
          (↑((wideSubquiverEquivSetTotal : (a : WideSubquiver (Generators G)) → Set (Quiver.Total (Generators G)))
                  (wideSubquiverSymmetrify T))ᶜ :
            Type u)),
        (F : (a : End (root' T)) → X)
            ((fun
                  (e :
                    (↑((wideSubquiverEquivSetTotal :
                              (a : WideSubquiver (Generators G)) → Set (Quiver.Total (Generators G)))
                            (wideSubquiverSymmetrify T))ᶜ :
                      Type u)) ↦
                (loopOfHom T (of (↑e : Quiver.Total (Generators G)).hom) : End (root' T)))
              a) =
          f a)
    (Functor.mapEnd (root' T) F')
refine_2 :
  ∀ (y : End (root' T) →* X),
    (fun (F : End (root' T) →* X) ↦
          ∀
            (a :
              (↑((wideSubquiverEquivSetTotal : (a : WideSubquiver (Generators G)) → Set (Quiver.Total (Generators G)))
                      (wideSubquiverSymmetrify T))ᶜ :
                Type u)),
            (F : (a : End (root' T)) → X)
                ((fun
                      (e :
                        (↑((wideSubquiverEquivSetTotal :
                                  (a : WideSubquiver (Generators G)) → Set (Quiver.Total (Generators G)))
                                (wideSubquiverSymmetrify T))ᶜ :
                          Type u)) ↦
                    (loopOfHom T (of (↑e : Quiver.Total (Generators G)).hom) : End (root' T)))
                  a) =
              f a)
        y →
      y = Functor.mapEnd (root' T) F'
⊢ ∃! F : End (root' T) →* X,
    ∀
      (a :
        (↑((wideSubquiverEquivSetTotal : (a : WideSubquiver (Generators G)) → Set (Quiver.Total (Generators G)))
                (wideSubquiverSymmetrify T))ᶜ :
          Type u)),
      (F : (a : End (root' T)) → X)
          ((fun
                (e :
                  (↑((wideSubquiverEquivSetTotal :
                            (a : WideSubquiver (Generators G)) → Set (Quiver.Total (Generators G)))
                          (wideSubquiverSymmetrify T))ᶜ :
                    Type u)) ↦
              (loopOfHom T (of (↑e : Quiver.Total (Generators G)).hom) : End (root' T)))
            a) =
        f a

Tactic:
refine ⟨F'.mapEnd _, ?_, ?_⟩