instance colimitSemiring : Semiring.{max v u} <| R.{v, u} F :=
  { (R.{v, u} F).str,
    AddCommMonCat.FilteredColimits.colimitAddCommMonoid
      (F ⋙ forget₂ SemiRingCat AddCommMonCat.{max v u}) with
    mul_zero := fun x => by
      refine Quot.inductionOn x ?_; clear x; intro x
      cases' x with j x
      erw [colimit_zero_eq _ j, colimit_mul_mk_eq _ ⟨j, _⟩ ⟨j, _⟩ j (𝟙 j) (𝟙 j)]
      rw [CategoryTheory.Functor.map_id]
      dsimp
      rw [mul_zero x]
      rfl
    zero_mul := fun x => by
      refine Quot.inductionOn x ?_; clear x; intro x
      cases' x with j x
      erw [colimit_zero_eq _ j, colimit_mul_mk_eq _ ⟨j, _⟩ ⟨j, _⟩ j (𝟙 j) (𝟙 j)]
      rw [CategoryTheory.Functor.map_id]
      dsimp
      rw [zero_mul x]
      rfl
    left_distrib := fun x y z => by
      refine Quot.induction_on₃ x y z ?_; clear x y z; intro x y z
      cases' x with j₁ x; cases' y with j₂ y; cases' z with j₃ z
      let k := IsFiltered.max₃ j₁ j₂ j₃
      let f := IsFiltered.firstToMax₃ j₁ j₂ j₃
      let g := IsFiltered.secondToMax₃ j₁ j₂ j₃
      let h := IsFiltered.thirdToMax₃ j₁ j₂ j₃
      erw [colimit_add_mk_eq _ ⟨j₂, _⟩ ⟨j₃, _⟩ k g h, colimit_mul_mk_eq _ ⟨j₁, _⟩ ⟨k, _⟩ k f (𝟙 k),
        colimit_mul_mk_eq _ ⟨j₁, _⟩ ⟨j₂, _⟩ k f g, colimit_mul_mk_eq _ ⟨j₁, _⟩ ⟨j₃, _⟩ k f h,
        colimit_add_mk_eq _ ⟨k, _⟩ ⟨k, _⟩ k (𝟙 k) (𝟙 k)]
      simp only [CategoryTheory.Functor.map_id, id_apply]
      erw [left_distrib (F.map f x) (F.map g y) (F.map h z)]
      rfl
    right_distrib := fun x y z => by
      refine Quot.induction_on₃ x y z ?_; clear x y z; intro x y z
      cases' x with j₁ x; cases' y with j₂ y; cases' z with j₃ z
      let k := IsFiltered.max₃ j₁ j₂ j₃
      let f := IsFiltered.firstToMax₃ j₁ j₂ j₃
      let g := IsFiltered.secondToMax₃ j₁ j₂ j₃
      let h := IsFiltered.thirdToMax₃ j₁ j₂ j₃
      erw [colimit_add_mk_eq _ ⟨j₁, _⟩ ⟨j₂, _⟩ k f g, colimit_mul_mk_eq _ ⟨k, _⟩ ⟨j₃, _⟩ k (𝟙 k) h,
        colimit_mul_mk_eq _ ⟨j₁, _⟩ ⟨j₃, _⟩ k f h, colimit_mul_mk_eq _ ⟨j₂, _⟩ ⟨j₃, _⟩ k g h,
        colimit_add_mk_eq _ ⟨k, _⟩ ⟨k, _⟩ k (𝟙 k) (𝟙 k)]
      

Proof state:
case mk.mk.mk
J : Type v
inst✝¹ : SmallCategory J
F : J ⥤ SemiRingCatMax
inst✝ : IsFiltered J
src✝¹ : Monoid ↑(R F) := (R F).str
src✝ : AddCommMonoid ↑(AddCommMonCat.FilteredColimits.M (F ⋙ forget₂ SemiRingCat AddCommMonCat)) :=
  AddCommMonCat.FilteredColimits.colimitAddCommMonoid (F ⋙ forget₂ SemiRingCat AddCommMonCat)
j₁ : J
x : ((F ⋙ forget₂ SemiRingCatMax MonCat) ⋙ forget MonCat).obj j₁
j₂ : J
y : ((F ⋙ forget₂ SemiRingCatMax MonCat) ⋙ forget MonCat).obj j₂
j₃ : J
z : ((F ⋙ forget₂ SemiRingCatMax MonCat) ⋙ forget MonCat).obj j₃
k : J := IsFiltered.max₃ j₁ j₂ j₃
f : j₁ ⟶ IsFiltered.max₃ j₁ j₂ j₃ := IsFiltered.firstToMax₃ j₁ j₂ j₃
g : j₂ ⟶ IsFiltered.max₃ j₁ j₂ j₃ := IsFiltered.secondToMax₃ j₁ j₂ j₃
h : j₃ ⟶ IsFiltered.max₃ j₁ j₂ j₃ := IsFiltered.thirdToMax₃ j₁ j₂ j₃
⊢ MonCat.FilteredColimits.M.mk (F ⋙ forget₂ SemiRingCatMax MonCat)
      { fst := k,
        snd :=
          ((F ⋙ forget₂ SemiRingCatMax MonCat).map (𝟙 k))
              { fst := k,
                  snd :=
                    (((F ⋙ forget₂ SemiRingCat AddCommMonCat) ⋙ forget₂ AddCommMonCat AddMonCat).map f)
                        { fst := j₁, snd := x }.snd +
                      (((F ⋙ forget₂ SemiRingCat AddCommMonCat) ⋙ forget₂ AddCommMonCat AddMonCat).map g)
                        { fst := j₂, snd := y }.snd }.snd *
            ((F ⋙ forget₂ SemiRingCatMax MonCat).map h) { fst := j₃, snd := z }.snd } =
    AddMonCat.FilteredColimits.M.mk ((F ⋙ forget₂ SemiRingCat AddCommMonCat) ⋙ forget₂ AddCommMonCat AddMonCat)
      { fst := k,
        snd :=
          (((F ⋙ forget₂ SemiRingCat AddCommMonCat) ⋙ forget₂ AddCommMonCat AddMonCat).map (𝟙 k))
              { fst := k,
                  snd :=
                    ((F ⋙ forget₂ SemiRingCatMax MonCat).map f) { fst := j₁, snd := x }.snd *
                      ((F ⋙ forget₂ SemiRingCatMax MonCat).map h) { fst := j₃, snd := z }.snd }.snd +
            (((F ⋙ forget₂ SemiRingCat AddCommMonCat) ⋙ forget₂ AddCommMonCat AddMonCat).map (𝟙 k))
              { fst := k,
                  snd :=
                    ((F ⋙ forget₂ SemiRingCatMax MonCat).map g) { fst := j₂, snd := y }.snd *
                      ((F ⋙ forget₂ SemiRingCatMax MonCat).map h) { fst := j₃, snd := z }.snd }.snd }

Next tactic:
simp only [CategoryTheory.Functor.map_id, id_apply]