∀ {ι : Type u_4} {G : ι → Type u_3} {H : Type u_1} {K : Type u_2} [inst : Monoid K] [inst_1 : (i : ι) → Monoid (G i)]
  [inst_2 : Monoid H] {φ : (i : ι) → H →* G i} (x : { f // ∀ (i : ι), (f.1 i).comp (φ i) = f.2 }) (i : ι),
  ((fun i => (Monoid.PushoutI.lift (↑x).1 (↑x).2 ⋯).comp (Monoid.PushoutI.of i),
              (Monoid.PushoutI.lift (↑x).1 (↑x).2 ⋯).comp (Monoid.PushoutI.base φ)).1
          i).comp
      (φ i) =
    (fun i => (Monoid.PushoutI.lift (↑x).1 (↑x).2 ⋯).comp (Monoid.PushoutI.of i),
        (Monoid.PushoutI.lift (↑x).1 (↑x).2 ⋯).comp (Monoid.PushoutI.base φ)).2