-- Module: Mathlib.Condensed.TopComparison
Goal:
C : Type u
inst : Category.{v, u} C
G : CategoryTheory.Functor.{v, w, u, w + 1} C TopCat
X : Type w'
inst_1 : TopologicalSpace X
inst_2 : ∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π], PreservesLimit (cospan π π) G
hq :
  ∀ (Z B : C) (π : Z ⟶ B) [EffectiveEpi π],
    IsQuotientMap (X := (↑(G.obj Z) : Type w)) (Y := (↑(G.obj B) : Type w))
      (⇑(ConcreteCategory.hom (FC := fun (X Y : TopCat.{w}) ↦ C((↑X : Type w), (↑Y : Type w))) (CC := TopCat.carrier)
            (G.map π)) :
        (a : (↑(G.obj Z) : Type w)) → (↑(G.obj B) : Type w))
Z B : C
π : Z ⟶ B
inst_3 : EffectiveEpi π
inst_4 : HasPullback π π
a b : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op B))
h :
  Eq (α :=
    (fun (X : Type (max w w')) ↦ X)
      (↑{x : (yonedaPresheaf G X).obj (op Z) |
            Eq (α := (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
              ((ConcreteCategory.hom (FC := TypeCat.Fun) ((yonedaPresheaf G X).map (Quiver.Hom.op (pullback.fst π π))) :
                    (a : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op Z))) →
                      (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                  x :
                (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
              ((ConcreteCategory.hom (FC := TypeCat.Fun) ((yonedaPresheaf G X).map (Quiver.Hom.op (pullback.snd π π))) :
                    (a : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op Z))) →
                      (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                  x :
                (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))} :
        Type (max w w')))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (mapToEqualizer (yonedaPresheaf G X) π (pullback.fst π π) (pullback.snd π π) ⋯) :
          (a : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op B))) →
            (fun (X : Type (max w w')) ↦ X)
              (↑{x : (yonedaPresheaf G X).obj (op Z) |
                    (ConcreteCategory.hom ((yonedaPresheaf G X).map (pullback.fst π π).op) :
                          (a : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op Z))) →
                            (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                        x =
                      (ConcreteCategory.hom ((yonedaPresheaf G X).map (pullback.snd π π).op) :
                          (a : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op Z))) →
                            (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                        x} :
                Type (max w w')))
        a :
      (fun (X : Type (max w w')) ↦ X)
        (↑{x : (yonedaPresheaf G X).obj (op Z) |
              Eq (α := (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                ((ConcreteCategory.hom (FC := TypeCat.Fun)
                        ((yonedaPresheaf G X).map (Quiver.Hom.op (pullback.fst π π))) :
                      (a : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op Z))) →
                        (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                    x :
                  (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                ((ConcreteCategory.hom (FC := TypeCat.Fun)
                        ((yonedaPresheaf G X).map (Quiver.Hom.op (pullback.snd π π))) :
                      (a : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op Z))) →
                        (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                    x :
                  (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))} :
          Type (max w w')))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (mapToEqualizer (yonedaPresheaf G X) π (pullback.fst π π) (pullback.snd π π) ⋯) :
          (a : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op B))) →
            (fun (X : Type (max w w')) ↦ X)
              (↑{x : (yonedaPresheaf G X).obj (op Z) |
                    (ConcreteCategory.hom ((yonedaPresheaf G X).map (pullback.fst π π).op) :
                          (a : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op Z))) →
                            (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                        x =
                      (ConcreteCategory.hom ((yonedaPresheaf G X).map (pullback.snd π π).op) :
                          (a : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op Z))) →
                            (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                        x} :
                Type (max w w')))
        b :
      (fun (X : Type (max w w')) ↦ X)
        (↑{x : (yonedaPresheaf G X).obj (op Z) |
              Eq (α := (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                ((ConcreteCategory.hom (FC := TypeCat.Fun)
                        ((yonedaPresheaf G X).map (Quiver.Hom.op (pullback.fst π π))) :
                      (a : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op Z))) →
                        (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                    x :
                  (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                ((ConcreteCategory.hom (FC := TypeCat.Fun)
                        ((yonedaPresheaf G X).map (Quiver.Hom.op (pullback.snd π π))) :
                      (a : (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op Z))) →
                        (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))
                    x :
                  (fun (X : Type (max w w')) ↦ X) ((yonedaPresheaf G X).obj (op (pullback π π))))} :
          Type (max w w')))
⊢ Eq (α := (↑(G.obj Z) : Type w) → X)
    (Function.comp (β := (↑(G.obj B) : Type w)) (⇑a : (a : (↑(G.obj B) : Type w)) → X)
      (⇑(TopCat.Hom.hom (G.map π)) : (a : (↑(G.obj Z) : Type w)) → (↑(G.obj B) : Type w)))
    (Function.comp (β := (↑(G.obj B) : Type w)) (⇑b : (a : (↑(G.obj B) : Type w)) → X)
      (⇑(TopCat.Hom.hom (G.map π)) : (a : (↑(G.obj Z) : Type w)) → (↑(G.obj B) : Type w)))

Tactic:
simp only [yonedaPresheaf, comp, Quiver.Hom.unop_op, TypeCat.Fun.coe_mk,
      Set.coe_setOf, mapToEqualizer, Set.mem_setOf_eq, ConcreteCategory.hom_ofHom, Subtype.mk.injEq,
      mk.injEq] at h

Premises used: Quiver.Hom.unop_op, CategoryTheory.regularTopology.mapToEqualizer, ContinuousMap.mk.injEq, ContinuousMap.comp, TypeCat.Fun.coe_mk, Set.mem_setOf_eq, Subtype.mk.injEq, ContinuousMap.yonedaPresheaf, Set.coe_setOf, CategoryTheory.ConcreteCategory.hom_ofHom