-- 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
pos :
  ∀ (h : p₀ ∈ (OpenNhds.inclusion p₀).obj (unop j)),
    CategoryStruct.comp (Y :=
        if p₀ ∈ (OpenNhds.inclusion p₀).obj (unop j) then colimit ((OpenNhds.inclusion p₀).op ⋙ X)
        else terminal.{u, v} C)
        (X.germ ((OpenNhds.inclusion p₀).obj (unop j)) p₀ ⋯ ≫ 𝟙 (X.stalk p₀) ≫ eqToHom ⋯)
        (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
neg :
  ∀ (h : p₀ ∉ (OpenNhds.inclusion p₀).obj (unop j)),
    CategoryStruct.comp (Y :=
        if p₀ ∈ (OpenNhds.inclusion p₀).obj (unop j) then colimit ((OpenNhds.inclusion p₀).op ⋙ X)
        else terminal.{u, v} C)
        (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 :=
      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

Tactic:
split_ifs with h