∀ (L : FirstOrder.Language) (ι : Type u_3) [inst : Preorder ι] (G : ι → Type u_4) [inst_1 : (i : ι) → L.Structure (G i)]
  (f : (i j : ι) → i ≤ j → L.Embedding (G i) (G j)) [inst_2 : IsDirectedOrder ι]
  [inst_3 : DirectedSystem G fun i j h => ⇑(f i j h)] [inst_4 : Nonempty ι] (G' : ι → Type u_5)
  [inst_5 : (i : ι) → L.Structure (G' i)] (f' : (i j : ι) → i ≤ j → L.Embedding (G' i) (G' j))
  (g : (i : ι) → L.Equiv (G i) (G' i)) [inst_6 : DirectedSystem G' fun i j h => ⇑(f' i j h)]
  (H_commuting : ∀ (i j : ι) (hij : i ≤ j) (x : G i), (g j) ((f i j hij) x) = (f' i j hij) ((g i) x)) {n : ℕ}
  (f_1 : L.Functions n) (x : Fin n → FirstOrder.Language.DirectLimit G f),
  (FirstOrder.Language.DirectLimit.lift L ι G f
          (fun i => (FirstOrder.Language.DirectLimit.of L ι G' f' i).comp (g i).toEmbedding) ⋯).toFun
      (FirstOrder.Language.Structure.funMap f_1 x) =
    FirstOrder.Language.Structure.funMap f_1
      ((FirstOrder.Language.DirectLimit.lift L ι G f
            (fun i => (FirstOrder.Language.DirectLimit.of L ι G' f' i).comp (g i).toEmbedding) ⋯).toFun ∘
        x)