-- Module: Mathlib.CategoryTheory.Join.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
a : D
a_1 : C
f : Hom.{v₁, v₂, u₁, u₂} (Join.right a) (left a_1)
a_2 : D
g : Hom.{v₁, v₂, u₁, u₂} (left a_1) (Join.right a_2)
a_3 : D
h : Hom.{v₁, v₂, u₁, u₂} (C := C) (Join.right a_2) (Join.right a_3)
right :
  Eq (α := ULift.{v₁, v₂} (a ⟶ a_3))
    {
      down :=
        ULift.down.{v₁, v₂}
            (match (motive :=
              (x x_1 x_2 : Join.{v₁, v₂, u₁, u₂} C D) →
                (match (motive := Join.{v₁, v₂, u₁, u₂} C D → Join.{v₁, v₂, u₁, u₂} C D → Type (max v₁ v₂)) x, x_1 with
                  | left x, left y => ULift.{v₂, v₁} (x ⟶ y)
                  | Join.right x, Join.right y => ULift.{v₁, v₂} (x ⟶ y)
                  | left a, Join.right a_4 => PUnit.{max (v₁ + 1) (v₂ + 1)}
                  | Join.right a, left a_4 => PEmpty.{max (v₁ + 1) (v₂ + 1)}) →
                  (match (motive := Join.{v₁, v₂, u₁, u₂} C D → Join.{v₁, v₂, u₁, u₂} C D → Type (max v₁ v₂)) x_1,
                      x_2 with
                    | left x, left y => ULift.{v₂, v₁} (x ⟶ y)
                    | Join.right x, Join.right y => ULift.{v₁, v₂} (x ⟶ y)
                    | left a, Join.right a_4 => PUnit.{max (v₁ + 1) (v₂ + 1)}
                    | Join.right a, left a_4 => PEmpty.{max (v₁ + 1) (v₂ + 1)}) →
                    match (motive := Join.{v₁, v₂, u₁, u₂} C D → Join.{v₁, v₂, u₁, u₂} C D → Type (max v₁ v₂)) x,
                      x_2 with
                    | left x, left y => ULift.{v₂, v₁} (x ⟶ y)
                    | Join.right x, Join.right y => ULift.{v₁, v₂} (x ⟶ y)
                    | left a, Join.right a_4 => PUnit.{max (v₁ + 1) (v₂ + 1)}
                    | Join.right a, left a_4 => PEmpty.{max (v₁ + 1) (v₂ + 1)})
              Join.right a, left a_1, Join.right a_2, f, g with
            | left _x, left _y, left _z, f, g => { down := f.down ≫ g.down }
            | left _x, left _y, Join.right _z, x, x_1 => PUnit.unit
            | left _x, Join.right _y, Join.right _z, x, x_1 => PUnit.unit
            | Join.right _x, Join.right _y, Join.right _z, f, g => { down := f.down ≫ g.down }) ≫
          h.down }
    (match (motive :=
      (x x_1 x_2 : Join.{v₁, v₂, u₁, u₂} C D) →
        (match (motive := Join.{v₁, v₂, u₁, u₂} C D → Join.{v₁, v₂, u₁, u₂} C D → Type (max v₁ v₂)) x, x_1 with
          | left x, left y => ULift.{v₂, v₁} (x ⟶ y)
          | Join.right x, Join.right y => ULift.{v₁, v₂} (x ⟶ y)
          | left a, Join.right a_4 => PUnit.{max (v₁ + 1) (v₂ + 1)}
          | Join.right a, left a_4 => PEmpty.{max (v₁ + 1) (v₂ + 1)}) →
          (match (motive := Join.{v₁, v₂, u₁, u₂} C D → Join.{v₁, v₂, u₁, u₂} C D → Type (max v₁ v₂)) x_1, x_2 with
            | left x, left y => ULift.{v₂, v₁} (x ⟶ y)
            | Join.right x, Join.right y => ULift.{v₁, v₂} (x ⟶ y)
            | left a, Join.right a_4 => PUnit.{max (v₁ + 1) (v₂ + 1)}
            | Join.right a, left a_4 => PEmpty.{max (v₁ + 1) (v₂ + 1)}) →
            match (motive := Join.{v₁, v₂, u₁, u₂} C D → Join.{v₁, v₂, u₁, u₂} C D → Type (max v₁ v₂)) x, x_2 with
            | left x, left y => ULift.{v₂, v₁} (x ⟶ y)
            | Join.right x, Join.right y => ULift.{v₁, v₂} (x ⟶ y)
            | left a, Join.right a_4 => PUnit.{max (v₁ + 1) (v₂ + 1)}
            | Join.right a, left a_4 => PEmpty.{max (v₁ + 1) (v₂ + 1)})
      Join.right a, left a_1, Join.right a_3, f, PUnit.unit with
    | left _x, left _y, left _z, f, g => { down := f.down ≫ g.down }
    | left _x, left _y, Join.right _z, x, x_1 => PUnit.unit
    | left _x, Join.right _y, Join.right _z, x, x_1 => PUnit.unit
    | Join.right _x, Join.right _y, Join.right _z, f, g => { down := f.down ≫ g.down })
⊢ comp (comp f g) h = comp f (comp g h)

Tactic:
simp only [Hom, comp, Category.assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.Join.comp, CategoryTheory.Join.Hom