-- Module: Mathlib.Topology.Gluing
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)
mp :
  D.Rel
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (X :=
          D.diagram.left (i, j)) (Y := of ((i : D.J) × (↑(D.U i) : Type u)))
          (NatTrans.app
              (Cocone.ι
                (Cofan.mk
                  (Cocone.pt (F := Discrete.functor D.diagram.right)
                    (colimit.cocone (Discrete.functor D.diagram.right)))
                  fun (i : (MultispanShape.prod D.J).L) ↦
                  D.diagram.fst i ≫
                    Cofan.inj (colimit.cocone (Discrete.functor D.diagram.right)) ((MultispanShape.prod D.J).fst i)))
              { as := (i, j) } ≫
            (sigmaIsoSigma D.U).hom) :
        (a : (↑(D.diagram.left (i, j)) : Type u)) → (↑(of ((i : D.J) × (↑(D.U i) : Type u))) : Type u))
      y)
    ((ConcreteCategory.hom (FC := fun (X Y : TopCat.{u}) ↦ C((↑X : Type u), (↑Y : Type u))) (CC := carrier) (X :=
          D.diagram.left (i, j)) (Y := of ((i : D.J) × (↑(D.U i) : Type u)))
          (NatTrans.app
              (Cocone.ι
                (Cofan.mk
                  (Cocone.pt (F := Discrete.functor D.diagram.right)
                    (colimit.cocone (Discrete.functor D.diagram.right)))
                  fun (i : (MultispanShape.prod D.J).L) ↦
                  D.diagram.snd i ≫
                    Cofan.inj (colimit.cocone (Discrete.functor D.diagram.right)) ((MultispanShape.prod D.J).snd i)))
              { as := (i, j) } ≫
            (sigmaIsoSigma D.U).hom) :
        (a : (↑(D.diagram.left (i, j)) : Type u)) → (↑(of ((i : D.J) × (↑(D.U i) : Type u))) : Type u))
      y)
⊢ 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:
erw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply, colimit.ι_desc_assoc,
      ← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply, colimit.ι_desc_assoc]

Premises used: CategoryTheory.ConcreteCategory.comp_apply, CategoryTheory.Limits.colimit.ι_desc_assoc