-- Module: Mathlib.Topology.Sheaves.Skyscraper
Goal:
X_1 : TopCat
p₀ : (↑X_1 : Type u)
inst : (U : Opens (↑X_1 : Type u)) → Decidable (p₀ ∈ U)
C : Type v
inst_1 : Category.{u, v} C
A : C
inst_2 : HasTerminal.{u, v} C
inst_3 : HasColimits.{u, v} C
X : Presheaf C X_1
j : (OpenNhds p₀)ᵒᵖ
w :
  CategoryStruct.comp (Y :=
      if p₀ ∈ (OpenNhds.inclusion p₀).obj (unop j) then colimit ((OpenNhds.inclusion p₀).op ⋙ X) else terminal.{u, v} C)
      (if h : p₀ ∈ (OpenNhds.inclusion p₀).obj (unop j) then
        X.germ ((OpenNhds.inclusion p₀).obj (unop j)) p₀ ⋯ ≫ 𝟙 (X.stalk p₀) ≫ eqToHom ⋯
      else
        IsTerminal.from
          (@Eq.rec _ _ (fun (x : C) (x_1 : terminal.{u, v} C = x) ↦ IsTerminal.{u, v} x) terminalIsTerminal
            (if p₀ ∈ (OpenNhds.inclusion p₀).obj (unop j) then colimit ((OpenNhds.inclusion p₀).op ⋙ X)
            else terminal.{u, v} C)
            ⋯)
          (X.obj (op ((OpenNhds.inclusion p₀).obj (unop j)))))
      (CategoryStruct.comp (Y :=
        colimit ((OpenNhds.inclusion p₀).op ⋙ skyscraperPresheaf p₀ (colimit ((OpenNhds.inclusion p₀).op ⋙ X))))
        (colimit.ι ((OpenNhds.inclusion p₀).op ⋙ skyscraperPresheaf p₀ (colimit ((OpenNhds.inclusion p₀).op ⋙ X))) j :
          Functor.obj ((OpenNhds.inclusion p₀).op ⋙ skyscraperPresheaf p₀ (colimit ((OpenNhds.inclusion p₀).op ⋙ X)))
              j ⟶
            colimit ((OpenNhds.inclusion p₀).op ⋙ skyscraperPresheaf p₀ (colimit ((OpenNhds.inclusion p₀).op ⋙ X))))
        (Iso.hom (skyscraperPresheafStalkOfSpecializes p₀ (colimit ((OpenNhds.inclusion p₀).op ⋙ X)) ⋯) :
          Presheaf.stalk (skyscraperPresheaf p₀ (colimit ((OpenNhds.inclusion p₀).op ⋙ X))) p₀ ⟶
            colimit ((OpenNhds.inclusion p₀).op ⋙ X))) =
    colimit.ι ((OpenNhds.inclusion p₀).op ⋙ X) j
⊢ CategoryStruct.comp (Y := colimit ((OpenNhds.inclusion p₀).op ⋙ X)) (colimit.ι ((OpenNhds.inclusion p₀).op ⋙ X) j)
      (CategoryStruct.comp (Y :=
        colimit ((OpenNhds.inclusion p₀).op ⋙ skyscraperPresheaf p₀ (colimit ((OpenNhds.inclusion p₀).op ⋙ X))))
        (colimMap
            ((OpenNhds.inclusion p₀).op.whiskerLeft
              {
                app := fun (U : (Opens (↑X_1 : Type u))ᵒᵖ) ↦
                  if h : p₀ ∈ unop U then X.germ (unop U) p₀ h ≫ 𝟙 (X.stalk p₀) ≫ eqToHom ⋯
                  else
                    IsTerminal.from
                      (@Eq.rec _ _ (fun (x : C) (x_1 : terminal.{u, v} C = x) ↦ IsTerminal.{u, v} x) terminalIsTerminal
                        (if p₀ ∈ unop U then colimit ((OpenNhds.inclusion p₀).op ⋙ X) else terminal.{u, v} C) ⋯)
                      (X.obj U),
                naturality := ⋯ : NatTrans X (skyscraperPresheaf p₀ (colimit ((OpenNhds.inclusion p₀).op ⋙ X))) }) :
          colimit ((OpenNhds.inclusion p₀).op ⋙ X) ⟶
            colimit ((OpenNhds.inclusion p₀).op ⋙ skyscraperPresheaf p₀ (colimit ((OpenNhds.inclusion p₀).op ⋙ X))))
        (Iso.hom (skyscraperPresheafStalkOfSpecializes p₀ (colimit ((OpenNhds.inclusion p₀).op ⋙ X)) ⋯) :
          Presheaf.stalk (skyscraperPresheaf p₀ (colimit ((OpenNhds.inclusion p₀).op ⋙ X))) p₀ ⟶
            colimit ((OpenNhds.inclusion p₀).op ⋙ X))) =
    colimit.ι ((OpenNhds.inclusion p₀).op ⋙ X) j ≫ 𝟙 (colimit ((OpenNhds.inclusion p₀).op ⋙ X))

Tactic:
simp only [Functor.comp_obj, Functor.op_obj, ι_colimMap_assoc, skyscraperPresheaf_obj,
      Functor.whiskerLeft_app, Category.comp_id]

Premises used: CategoryTheory.Functor.comp_obj, skyscraperPresheaf_obj, CategoryTheory.Limits.ι_colimMap_assoc, CategoryTheory.Functor.whiskerLeft_app, CategoryTheory.Category.comp_id, CategoryTheory.Functor.op_obj