-- Lean 4 proof step from Mathlib.Topology.Gluing (goal)
Goal:
D : GlueData
i_1 j_1 : D.J
x : (↑(D.U i_1) : Type u)
y_1 : (↑(D.U j_1) : Type u)
h :
  Eq (α := (↑D.glued : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued)
            (CategoryStruct.comp (Y := ∐ D.diagram.right) (Sigma.ι D.diagram.right i_1)
              (Multicoequalizer.sigmaπ D.diagram)) :
          (a : (↑(D.U i_1) : Type u)) → (↑D.glued : Type u))
        x :
      (↑D.glued : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (Y :=
            D.glued)
            (CategoryStruct.comp (Y := ∐ D.diagram.right) (Sigma.ι D.diagram.right j_1)
              (Multicoequalizer.sigmaπ D.diagram)) :
          (a : (↑(D.U j_1) : Type u)) → (↑D.glued : Type u))
        y_1 :
      (↑D.glued : Type u))
i j : D.J
y : (↑(D.diagram.left (i, j)) : Type u)
⊢ D.Rel
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
          (sigmaIsoSigma D.U).hom :
        (a : (↑(∐ D.U) : Type u)) → (↑(of ((i : D.J) × (↑(D.U i) : Type u))) : Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (X :=
              ∐ D.diagram.left) (Y := ∐ D.diagram.right)
              (D.diagram.fstSigmaMapOfIsColimit (colimit.cocone (Discrete.functor D.diagram.right))
                (colimit.isColimit (Discrete.functor D.diagram.left))) :
            (a : (↑(∐ D.diagram.left) : Type u)) → (↑(∐ D.diagram.right) : Type u))
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                  (sigmaIsoSigma D.diagram.left).inv :
                (a : (↑(of ((i : (MultispanShape.prod D.J).L) × (↑(D.diagram.left i) : Type u))) : Type u)) →
                  (↑(∐ D.diagram.left) : Type u))
              (Sigma.mk (β := fun (i : (MultispanShape.prod D.J).L) ↦ (↑(D.diagram.left i) : Type u)) (i, j) y) :
            (↑(∐ D.diagram.left) : Type u)) :
        (↑(∐ D.diagram.right) : Type u)))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
          (sigmaIsoSigma D.U).hom :
        (a : (↑(∐ D.U) : Type u)) → (↑(of ((i : D.J) × (↑(D.U i) : Type u))) : Type u))
      ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (X :=
              ∐ D.diagram.left) (Y := ∐ D.diagram.right)
              (D.diagram.sndSigmaMapOfIsColimit (colimit.cocone (Discrete.functor D.diagram.right))
                (colimit.isColimit (Discrete.functor D.diagram.left))) :
            (a : (↑(∐ D.diagram.left) : Type u)) → (↑(∐ D.diagram.right) : Type u))
          ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
                  (sigmaIsoSigma D.diagram.left).inv :
                (a : (↑(of ((i : (MultispanShape.prod D.J).L) × (↑(D.diagram.left i) : Type u))) : Type u)) →
                  (↑(∐ D.diagram.left) : Type u))
              (Sigma.mk (β := fun (i : (MultispanShape.prod D.J).L) ↦ (↑(D.diagram.left i) : Type u)) (i, j) y) :
            (↑(∐ D.diagram.left) : Type u)) :
        (↑(∐ D.diagram.right) : Type u)))
have mp : D.Rel
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
        (sigmaIsoSigma D.U).hom :
      (a : (↑(∐ D.U) : Type u)) → (↑(of ((i : D.J) × (↑(D.U i) : Type u))) : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (X :=
            ∐ D.diagram.left) (Y := ∐ D.diagram.right)
            (D.diagram.fstSigmaMapOfIsColimit (colimit.cocone (Discrete.functor D.diagram.right))
              (colimit.isColimit (Discrete.functor D.diagram.left))) :
          (a : (↑(∐ D.diagram.left) : Type u)) → (↑(∐ D.diagram.right) : Type u))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (X :=
                D.diagram.left (i, j)) (Y := ∐ D.diagram.left) (Sigma.ι D.diagram.left (i, j)) :
              (a : (↑(D.diagram.left (i, j)) : Type u)) → (↑(∐ D.diagram.left) : Type u))
            y :
          (↑(∐ D.diagram.left) : Type u)) :
      (↑(∐ D.diagram.right) : Type u)))
  ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier)
        (sigmaIsoSigma D.U).hom :
      (a : (↑(∐ D.U) : Type u)) → (↑(of ((i : D.J) × (↑(D.U i) : Type u))) : Type u))
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (X :=
            ∐ D.diagram.left) (Y := ∐ D.diagram.right)
            (D.diagram.sndSigmaMapOfIsColimit (colimit.cocone (Discrete.functor D.diagram.right))
              (colimit.isColimit (Discrete.functor D.diagram.left))) :
          (a : (↑(∐ D.diagram.left) : Type u)) → (↑(∐ D.diagram.right) : Type u))
        ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (X :=
                D.diagram.left (i, j)) (Y := ∐ D.diagram.left) (Sigma.ι D.diagram.left (i, j)) :
              (a : (↑(D.diagram.left (i, j)) : Type u)) → (↑(∐ D.diagram.left) : Type u))
            y :
          (↑(∐ D.diagram.left) : Type u)) :
      (↑(∐ D.diagram.right) : Type u)))
Tactic: rw [sigmaIsoSigma_inv_apply]