-- Module: Mathlib.Algebra.Category.Ring.FilteredColimits
Goal:
J : Type v
inst : SmallCategory J
F : J ⥤ SemiRingCat
inst_1 : IsFiltered J
t : Cocone F
x_1 : (↑(R.{v, u} F) : Type (max v u))
j : J
x :
  ((F ⋙
          forget₂ SemiRingCat (FC := fun (R S : SemiRingCat) ↦ (↑R : Type (max u v)) →+* (↑S : Type (max u v))) (CC :=
            carrier) MonCat (FD := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v))) (CD :=
            MonCat.carrier)) ⋙
        forget MonCat (FC := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v))) (CC :=
          MonCat.carrier)).obj
    j
mk :
  Eq (α := (↑t.pt : Type (max u v)))
    ((ConcreteCategory.hom (FC := fun (R S : SemiRingCat) ↦ (↑R : Type (max u v)) →+* (↑S : Type (max u v))) (CC :=
            carrier) (t.ι.app j) :
          (a : (↑(F.obj j) : Type (max u v))) → (↑(((Functor.const J).obj t.pt).obj j) : Type (max u v)))
        x :
      (↑(Functor.obj (Functor.obj (Functor.const J) t.pt) j) : Type (max u v)))
    ((descAddMonoidHom t : (a : (↑(R F) : Type (max v u))) → (↑t.pt : Type (max u v)))
        (Quot.mk
          ((F ⋙
                forget₂ SemiRingCat (FC := fun (R S : SemiRingCat) ↦ (↑R : Type (max u v)) →+* (↑S : Type (max u v)))
                  (CC := carrier) MonCat (FD := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v)))
                  (CD := MonCat.carrier)) ⋙
              forget MonCat (FC := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v))) (CC :=
                MonCat.carrier)).ColimitTypeRel
          (Sigma.mk (β := fun (j : J) ↦
            ((F ⋙
                    forget₂ SemiRingCat (FC := fun (R S : SemiRingCat) ↦
                      (↑R : Type (max u v)) →+* (↑S : Type (max u v))) (CC := carrier) MonCat (FD :=
                      fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v))) (CD := MonCat.carrier)) ⋙
                  forget MonCat (FC := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v))) (CC :=
                    MonCat.carrier)).obj
              j)
            j x)) :
      (↑t.pt : Type (max u v)))
mk₁ mk₂ :
  Eq (α := (↑t.pt : Type (max u v)))
    ((ConcreteCategory.hom (FC := fun (R S : SemiRingCat) ↦ (↑R : Type (max u v)) →+* (↑S : Type (max u v))) (CC :=
            carrier) (t.ι.app j) :
          (a : (↑(F.obj j) : Type (max u v))) → (↑(((Functor.const J).obj t.pt).obj j) : Type (max u v)))
        x :
      (↑(Functor.obj (Functor.obj (Functor.const J) t.pt) j) : Type (max u v)))
    ((ConcreteCategory.hom (FC := fun (R S : SemiRingCat) ↦ (↑R : Type (max u v)) →+* (↑S : Type (max u v))) (CC :=
            carrier) (t.ι.app j) :
          (a : (↑(F.obj j) : Type (max u v))) → (↑(((Functor.const J).obj t.pt).obj j) : Type (max u v)))
        x :
      (↑(Functor.obj (Functor.obj (Functor.const J) t.pt) j) : Type (max u v)))
⊢ Eq (α := (↑t.pt : Type (max u v)))
    ((descMonoidHom t : (a : (↑(R F) : Type (max v u))) → (↑t.pt : Type (max u v)))
        (Quot.mk
          ((F ⋙
                forget₂ SemiRingCat (FC := fun (R S : SemiRingCat) ↦ (↑R : Type (max u v)) →+* (↑S : Type (max u v)))
                  (CC := carrier) MonCat (FD := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v)))
                  (CD := MonCat.carrier)) ⋙
              forget MonCat (FC := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v))) (CC :=
                MonCat.carrier)).ColimitTypeRel
          (Sigma.mk (β := fun (j : J) ↦
            ((F ⋙
                    forget₂ SemiRingCat (FC := fun (R S : SemiRingCat) ↦
                      (↑R : Type (max u v)) →+* (↑S : Type (max u v))) (CC := carrier) MonCat (FD :=
                      fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v))) (CD := MonCat.carrier)) ⋙
                  forget MonCat (FC := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v))) (CC :=
                    MonCat.carrier)).obj
              j)
            j x)) :
      (↑t.pt : Type (max u v)))
    ((descAddMonoidHom t : (a : (↑(R F) : Type (max v u))) → (↑t.pt : Type (max u v)))
        (Quot.mk
          ((F ⋙
                forget₂ SemiRingCat (FC := fun (R S : SemiRingCat) ↦ (↑R : Type (max u v)) →+* (↑S : Type (max u v)))
                  (CC := carrier) MonCat (FD := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v)))
                  (CD := MonCat.carrier)) ⋙
              forget MonCat (FC := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v))) (CC :=
                MonCat.carrier)).ColimitTypeRel
          (Sigma.mk (β := fun (j : J) ↦
            ((F ⋙
                    forget₂ SemiRingCat (FC := fun (R S : SemiRingCat) ↦
                      (↑R : Type (max u v)) →+* (↑S : Type (max u v))) (CC := carrier) MonCat (FD :=
                      fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v))) (CD := MonCat.carrier)) ⋙
                  forget MonCat (FC := fun (x1 x2 : MonCat) ↦ (↑x1 : Type (max u v)) →* (↑x2 : Type (max u v))) (CC :=
                    MonCat.carrier)).obj
              j)
            j x)) :
      (↑t.pt : Type (max u v)))

Tactic:
rw [descMonoidHom_quotMk t x, descAddMonoidHom_quotMk t x]

Premises used: SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descMonoidHom_quotMk, SemiRingCat.FilteredColimits.colimitCoconeIsColimit.descAddMonoidHom_quotMk