-- Module: Mathlib.Algebra.Category.Ring.Adjunctions
Goal:
R : CommRingCat
S : Under R
dsimp :
  CategoryStruct.comp (Y :=
      Functor.obj
        (forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := carrier)
          CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD := CommMonCat.carrier))
        (of (MonoidAlgebra (↑R : Type u) (↑S.right : Type u))))
      (CommMonCat.ofHom (MonoidAlgebra.of (↑R : Type u) (↑S.right : Type u)))
      (CommMonCat.ofHom
        (↑(MonoidAlgebra.liftNCRingHom (Hom.hom S.hom) (MonoidHom.id (↑S.right : Type u)) ⋯) :
          MonoidAlgebra (↑R : Type u) (↑S.right : Type u) →* (↑S.right : Type u))) =
    𝟙
      (Functor.obj
        (forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := carrier)
          CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD := CommMonCat.carrier))
        S.right)
⊢ Eq (α :=
    Functor.obj
        (Under.forget R ⋙
          forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := carrier)
            CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
            CommMonCat.carrier))
        S ⟶
      Functor.obj
        (Under.forget R ⋙
          forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := carrier)
            CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
            CommMonCat.carrier))
        S)
    (CategoryStruct.comp (Y :=
        Functor.obj
          (Under.forget R ⋙
            forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := carrier)
              CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
              CommMonCat.carrier))
          (R.monoidAlgebra.obj
            (Functor.obj
              (Under.forget R ⋙
                forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC :=
                  carrier) CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
                  CommMonCat.carrier))
              S)))
        (NatTrans.app
          { app := fun (G : CommMonCat.{u}) ↦ CommMonCat.ofHom (MonoidAlgebra.of (↑R : Type u) (↑G : Type u)),
            naturality := ⋯ }
          (Functor.obj
            (Under.forget R ⋙
              forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := carrier)
                CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
                CommMonCat.carrier))
            S))
        (CategoryTheory.Functor.map
          (Under.forget R ⋙
            forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := carrier)
              CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
              CommMonCat.carrier))
          (NatTrans.app
            {
              app := fun (S : Under R) ↦
                Under.homMk
                  (ofHom
                    (MonoidAlgebra.liftNCRingHom (Hom.hom S.hom)
                      (MonoidHom.id
                        (↑(Functor.obj
                              (Under.forget R ⋙
                                forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦
                                  (↑R : Type u) →+* (↑S : Type u)) (CC := carrier) CommMonCat (FD :=
                                  fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
                                  CommMonCat.carrier))
                              S) :
                          Type u))
                      ⋯))
                  ⋯,
              naturality := ⋯ }
            S)) :
      Functor.obj
          (Under.forget R ⋙
            forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := carrier)
              CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
              CommMonCat.carrier))
          S ⟶
        Functor.obj
          (Under.forget R ⋙
            forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := carrier)
              CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
              CommMonCat.carrier))
          S)
    (𝟙
        (Functor.obj
          (Under.forget R ⋙
            forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := carrier)
              CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
              CommMonCat.carrier))
          S) :
      Functor.obj
          (Under.forget R ⋙
            forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := carrier)
              CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
              CommMonCat.carrier))
          S ⟶
        Functor.obj
          (Under.forget R ⋙
            forget₂ CommRingCat (FC := fun (R S : CommRingCat.{u}) ↦ (↑R : Type u) →+* (↑S : Type u)) (CC := carrier)
              CommMonCat (FD := fun (x1 x2 : CommMonCat.{u}) ↦ (↑x1 : Type u) →* (↑x2 : Type u)) (CD :=
              CommMonCat.carrier))
          S)

Tactic:
dsimp