-- Module: Mathlib.AlgebraicGeometry.GammaSpecAdjunction
Goal:
X : LocallyRingedSpace
r_1 : (↑(Γ.obj (op X)) : Type u)
r s : (InducedCategory (C := (↑(Γ.obj (op X)) : Type u)) (Opens (PrimeSpectrum (↑(Γ.obj (op X)) : Type u))) basicOpen)ᵒᵖ
f : r ⟶ s
a :
  Eq (α :=
    CommRingCat.of (↑(Γ.obj (op X)) : Type u) ⟶
      Functor.obj
        (Functor.op (inducedFunctor basicOpen) ⋙
          (Functor.obj (TopCat.Sheaf.pushforward CommRingCat X.toΓSpecBase) X.𝒪).obj (P :=
            Presheaf.IsSheaf (Opens.grothendieckTopology (↑(Spec.topObj (Γ.obj (op X))) : Type u))))
        s)
    (CategoryStruct.comp (Y :=
      Functor.obj
        (Functor.op (inducedFunctor basicOpen) ⋙
          ObjectProperty.FullSubcategory.obj (P :=
            Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u)))
            (structureSheaf (↑(Γ.obj (op X)) : Type u)))
        s)
      (CommRingCat.ofHom
          (algebraMap (↑(Γ.obj (op X)) : Type u)
            (↑(Functor.obj (C := (Opens (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u))ᵒᵖ)
                  (ObjectProperty.FullSubcategory.obj (P :=
                    Presheaf.IsSheaf
                      (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u)))
                    (structureSheaf (↑(Γ.obj (op X)) : Type u)))
                  (op (basicOpen (unop r)))) :
              Type u)) ≫
        CategoryTheory.Functor.map
          (Functor.op (inducedFunctor basicOpen) ⋙
            ObjectProperty.FullSubcategory.obj (P :=
              Presheaf.IsSheaf (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u)))
              (structureSheaf (↑(Γ.obj (op X)) : Type u)))
          f)
      (X.toΓSpecCApp (unop s)))
    (CategoryStruct.comp (Y :=
      Functor.obj
        (Functor.op (inducedFunctor basicOpen) ⋙
          (Functor.obj (TopCat.Sheaf.pushforward CommRingCat X.toΓSpecBase) X.𝒪).obj (P :=
            Presheaf.IsSheaf (Opens.grothendieckTopology (↑(Spec.topObj (Γ.obj (op X))) : Type u))))
        r)
      (CommRingCat.ofHom
          (algebraMap (↑(Γ.obj (op X)) : Type u)
            (↑(Functor.obj (C := (Opens (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u))ᵒᵖ)
                  (ObjectProperty.FullSubcategory.obj (P :=
                    Presheaf.IsSheaf
                      (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u)))
                    (structureSheaf (↑(Γ.obj (op X)) : Type u)))
                  (op (basicOpen (unop r)))) :
              Type u)) ≫
        X.toΓSpecCApp (unop r))
      (CategoryTheory.Functor.map
        (Functor.op (inducedFunctor basicOpen) ⋙
          (Functor.obj (TopCat.Sheaf.pushforward CommRingCat X.toΓSpecBase) X.𝒪).obj (P :=
            Presheaf.IsSheaf (Opens.grothendieckTopology (↑(Spec.topObj (Γ.obj (op X))) : Type u))))
        f))
⊢ Eq (α :=
    CommRingCat.of (↑(Γ.obj (op X)) : Type u) ⟶
      Functor.obj
        (Functor.op (inducedFunctor basicOpen) ⋙
          (Functor.obj (TopCat.Sheaf.pushforward CommRingCat X.toΓSpecBase) X.𝒪).obj (P :=
            Presheaf.IsSheaf (Opens.grothendieckTopology (↑(Spec.topObj (Γ.obj (op X))) : Type u))))
        s)
    (CategoryStruct.comp (Y :=
      CommRingCat.of
        (↑(Functor.obj (C := (Opens (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u))ᵒᵖ)
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf
                  (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u)))
                (structureSheaf (↑(Γ.obj (op X)) : Type u)))
              (op (basicOpen (unop r)))) :
          Type u))
      (CommRingCat.ofHom
          (algebraMap (↑(Γ.obj (op X)) : Type u)
            (↑(Functor.obj (C := (Opens (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u))ᵒᵖ)
                  (ObjectProperty.FullSubcategory.obj (P :=
                    Presheaf.IsSheaf
                      (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u)))
                    (structureSheaf (↑(Γ.obj (op X)) : Type u)))
                  (op (basicOpen (unop r)))) :
              Type u)) :
        CommRingCat.of (↑(Γ.obj (op X)) : Type u) ⟶
          CommRingCat.of
            (↑(Functor.obj (C := (Opens (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u))ᵒᵖ)
                  (ObjectProperty.FullSubcategory.obj (P :=
                    Presheaf.IsSheaf
                      (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u)))
                    (structureSheaf (↑(Γ.obj (op X)) : Type u)))
                  (op (basicOpen (unop r)))) :
              Type u))
      (CategoryTheory.Functor.map
            (Functor.op (inducedFunctor basicOpen) ⋙
              ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf
                  (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u)))
                (structureSheaf (↑(Γ.obj (op X)) : Type u)))
            f ≫
          X.toΓSpecCApp (unop s) :
        Functor.obj
            (Functor.op (inducedFunctor basicOpen) ⋙
              ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf
                  (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u)))
                (structureSheaf (↑(Γ.obj (op X)) : Type u)))
            r ⟶
          Functor.obj
            (Functor.op (inducedFunctor basicOpen) ⋙
              (Functor.obj (TopCat.Sheaf.pushforward CommRingCat X.toΓSpecBase) X.𝒪).obj (P :=
                Presheaf.IsSheaf (Opens.grothendieckTopology (↑(Spec.topObj (Γ.obj (op X))) : Type u))))
            s))
    (CategoryStruct.comp (Y :=
      CommRingCat.of
        (↑(Functor.obj (C := (Opens (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u))ᵒᵖ)
              (ObjectProperty.FullSubcategory.obj (P :=
                Presheaf.IsSheaf
                  (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u)))
                (structureSheaf (↑(Γ.obj (op X)) : Type u)))
              (op (basicOpen (unop r)))) :
          Type u))
      (CommRingCat.ofHom
        (algebraMap (↑(Γ.obj (op X)) : Type u)
          (↑(Functor.obj (C := (Opens (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u))ᵒᵖ)
                (ObjectProperty.FullSubcategory.obj (P :=
                  Presheaf.IsSheaf
                    (Opens.grothendieckTopology (↑(PrimeSpectrum.Top (↑(Γ.obj (op X)) : Type u)) : Type u)))
                  (structureSheaf (↑(Γ.obj (op X)) : Type u)))
                (op (basicOpen (unop r)))) :
            Type u)))
      (X.toΓSpecCApp (unop r) ≫
        CategoryTheory.Functor.map
          (Functor.op (inducedFunctor basicOpen) ⋙
            (Functor.obj (TopCat.Sheaf.pushforward CommRingCat X.toΓSpecBase) X.𝒪).obj (P :=
              Presheaf.IsSheaf (Opens.grothendieckTopology (↑(Spec.topObj (Γ.obj (op X))) : Type u))))
          f))

Tactic:
simp only [← Category.assoc]

Premises used: CategoryTheory.Category.assoc