-- Lean 4 proof step from Mathlib.CategoryTheory.Bicategory.Grothendieck (goal)
Goal:
𝒮 : Type u₁
inst : Category.{v₁, u₁} 𝒮
F G H : LocallyDiscrete 𝒮ᵒᵖ ⥤ᵖ Cat
α : F ⟶ G
a : ∫ᶜ F
hfg₁ :
  { base := Hom.base (𝟙 a),
        fiber :=
          Functor.map (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) (Hom.fiber (𝟙 a)) ≫
            NatTrans.app (Cat.Hom₂.toNatTrans (Iso.hom (α.naturality (Hom.base (𝟙 a)).op.toLoc))) a.fiber :
        { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
              ∫ᶜ G }.Hom
          { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
            ∫ᶜ G } }.base =
    Hom.base
      (𝟙 { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber })
⊢ { base := Hom.base (𝟙 a),
      fiber :=
        Functor.map (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) (Hom.fiber (𝟙 a)) ≫
          NatTrans.app (Cat.Hom₂.toNatTrans (Iso.hom (α.naturality (Hom.base (𝟙 a)).op.toLoc))) a.fiber :
      { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
            ∫ᶜ G }.Hom
        { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
          ∫ᶜ G } } =
    𝟙 { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber }
have hfg₂ : Eq (α :=
  { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
        ∫ᶜ G }.fiber ⟶
    obj
      (Cat.Hom.toFunctor
        (G.map
          { base := Hom.base (𝟙 a),
                  fiber :=
                    Functor.map (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ }))
                        (Hom.fiber (𝟙 a)) ≫
                      NatTrans.app (Cat.Hom₂.toNatTrans (Iso.hom (α.naturality (Hom.base (𝟙 a)).op.toLoc))) a.fiber :
                  { base := a.base,
                        fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
                        ∫ᶜ G }.Hom
                    { base := a.base,
                      fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
                      ∫ᶜ G } }.base.op.toLoc))
      { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
          ∫ᶜ G }.fiber)
  ({ base := Hom.base (𝟙 a),
        fiber :=
          Functor.map (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) (Hom.fiber (𝟙 a)) ≫
            NatTrans.app (Cat.Hom₂.toNatTrans (Iso.hom (α.naturality (Hom.base (𝟙 a)).op.toLoc))) a.fiber }.fiber :
    { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
          ∫ᶜ G }.fiber ⟶
      obj
        (Cat.Hom.toFunctor
          (G.map
            { base := Hom.base (𝟙 a),
                    fiber :=
                      Functor.map (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ }))
                          (Hom.fiber (𝟙 a)) ≫
                        NatTrans.app (Cat.Hom₂.toNatTrans (Iso.hom (α.naturality (Hom.base (𝟙 a)).op.toLoc))) a.fiber :
                    { base := a.base,
                          fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
                          ∫ᶜ G }.Hom
                      { base := a.base,
                        fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
                        ∫ᶜ G } }.base.op.toLoc))
        { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
            ∫ᶜ G }.fiber)
  (Hom.fiber
        (𝟙
          { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
            ∫ᶜ G }) ≫
      eqToHom sorry :
    { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
          ∫ᶜ G }.fiber ⟶
      obj
        (Cat.Hom.toFunctor
          (G.map
            { base := Hom.base (𝟙 a),
                    fiber :=
                      Functor.map (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ }))
                          (Hom.fiber (𝟙 a)) ≫
                        NatTrans.app (Cat.Hom₂.toNatTrans (Iso.hom (α.naturality (Hom.base (𝟙 a)).op.toLoc))) a.fiber :
                    { base := a.base,
                          fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
                          ∫ᶜ G }.Hom
                      { base := a.base,
                        fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
                        ∫ᶜ G } }.base.op.toLoc))
        { base := a.base, fiber := obj (Cat.Hom.toFunctor (α.app { as := op a.base : LocallyDiscrete 𝒮ᵒᵖ })) a.fiber :
            ∫ᶜ G }.fiber)
Tactic: ext1