-- Module: Mathlib.Algebra.Category.ModuleCat.Limits
Goal:
R : Type u
inst : Ring R
J : Type v
inst_1 : Category.{t, v} J
F : J ⥤ ModuleCat.{w, u} R
inst_2 :
  Small.{w, max v w}
    (↑(F ⋙
            forget (ModuleCat.{w, u} R) (FC := fun (x1 x2 : ModuleCat.{w, u} R) ↦ (↑x1 : Type w) →ₗ[R] (↑x2 : Type w))
              (CC := carrier)).sections :
      Type (max v w))
j : J
x x_1 :
  (Types.Small.limitCone
      (F ⋙
        forget (ModuleCat.{w, u} R) (FC := fun (x1 x2 : ModuleCat.{w, u} R) ↦ (↑x1 : Type w) →ₗ[R] (↑x2 : Type w))
          (CC := carrier))).pt
⊢ (↑(HAdd.hAdd (α :=
            (↑(F ⋙
                    forget (ModuleCat.{w, u} R) (FC := fun (x1 x2 : ModuleCat.{w, u} R) ↦
                      (↑x1 : Type w) →ₗ[R] (↑x2 : Type w)) (CC := carrier)).sections :
              Type (max w v)))
            (β :=
            (↑(F ⋙
                    forget (ModuleCat.{w, u} R) (FC := fun (x1 x2 : ModuleCat.{w, u} R) ↦
                      (↑x1 : Type w) →ₗ[R] (↑x2 : Type w)) (CC := carrier)).sections :
              Type (max w v)))
            ((Equiv.symm
                    (equivShrink
                      (↑(F ⋙
                              forget (ModuleCat.{w, u} R) (FC := fun (x1 x2 : ModuleCat.{w, u} R) ↦
                                (↑x1 : Type w) →ₗ[R] (↑x2 : Type w)) (CC := carrier)).sections :
                        Type (max w v))) :
                  (a : Shrink.{w, max v w} (↑(F ⋙ forget (ModuleCat R)).sections : Type (max w v))) →
                    (↑(F ⋙ forget (ModuleCat R)).sections : Type (max w v)))
                x :
              (↑(F ⋙
                      forget (ModuleCat.{w, u} R) (FC := fun (x1 x2 : ModuleCat.{w, u} R) ↦
                        (↑x1 : Type w) →ₗ[R] (↑x2 : Type w)) (CC := carrier)).sections :
                Type (max w v)))
            ((Equiv.symm
                    (equivShrink
                      (↑(F ⋙
                              forget (ModuleCat.{w, u} R) (FC := fun (x1 x2 : ModuleCat.{w, u} R) ↦
                                (↑x1 : Type w) →ₗ[R] (↑x2 : Type w)) (CC := carrier)).sections :
                        Type (max w v))) :
                  (a : Shrink.{w, max v w} (↑(F ⋙ forget (ModuleCat R)).sections : Type (max w v))) →
                    (↑(F ⋙ forget (ModuleCat R)).sections : Type (max w v)))
                x_1 :
              (↑(F ⋙
                      forget (ModuleCat.{w, u} R) (FC := fun (x1 x2 : ModuleCat.{w, u} R) ↦
                        (↑x1 : Type w) →ₗ[R] (↑x2 : Type w)) (CC := carrier)).sections :
                Type (max w v)))) :
        (j : J) → ToType (F.obj j))
      j =
    HAdd.hAdd (α :=
      ToType (FC := fun (x1 x2 : ModuleCat.{w, u} R) ↦ (↑x1 : Type w) →ₗ[R] (↑x2 : Type w)) (CC := carrier) (F.obj j))
      ((↑((Equiv.symm
                  (equivShrink
                    (↑(F ⋙
                            forget (ModuleCat.{w, u} R) (FC := fun (x1 x2 : ModuleCat.{w, u} R) ↦
                              (↑x1 : Type w) →ₗ[R] (↑x2 : Type w)) (CC := carrier)).sections :
                      Type (max w v))) :
                (a : Shrink.{w, max w v} (↑(F ⋙ forget (ModuleCat R)).sections : Type (max w v))) →
                  (↑(F ⋙ forget (ModuleCat R)).sections : Type (max w v)))
              x) :
          (j : J) → ToType (F.obj j))
        j)
      ((↑((Equiv.symm
                  (equivShrink
                    (↑(F ⋙
                            forget (ModuleCat.{w, u} R) (FC := fun (x1 x2 : ModuleCat.{w, u} R) ↦
                              (↑x1 : Type w) →ₗ[R] (↑x2 : Type w)) (CC := carrier)).sections :
                      Type (max w v))) :
                (a : Shrink.{w, max w v} (↑(F ⋙ forget (ModuleCat R)).sections : Type (max w v))) →
                  (↑(F ⋙ forget (ModuleCat R)).sections : Type (max w v)))
              x_1) :
          (j : J) → ToType (F.obj j))
        j)

Tactic:
rfl