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+import Mathlib
+import Algebra.Terms
+import Algebra.Derivations
+
+open CategoryTheory
+
+namespace NominalAlgebra
+
+section AlgebraDefinition
+
+variable (𝔸 : Type) [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
+variable (sig : Signature)
+
+structure Algebra where
+  carrier : ℕ → Nom 𝔸
+  pure (f : sig.pure) {m : ℕ} :
+    (sig.pureArgs f → carrier m) →ₑ[𝔸] carrier (m + sig.pureDepth f)
+  free (f : sig.free) {m : ℕ} :
+    (𝔸 × (sig.freeArgs f → carrier m)) →ₑ[𝔸] carrier (m + sig.freeDepth f)
+  bound (f : sig.bound) {m : ℕ} :
+    NameAbstraction 𝔸 (sig.boundArgs f → carrier m) →ₑ[𝔸] carrier (m + sig.boundDepth f)
+
+instance : CoeFun (Algebra 𝔸 sig) (fun _ => ℕ → Nom 𝔸) where
+  coe A := A.carrier
+
+@[ext]
+structure Algebra.Hom (A B : Algebra 𝔸 sig) where
+  toFun : (n : ℕ) → (A n →ₑ[𝔸] B n)
+  preserve_pure : ∀ (f : sig.pure) {m : ℕ} (args : sig.pureArgs f → A m),
+    toFun (m + sig.pureDepth f) (A.pure f args) = B.pure f (toFun m ∘ args)
+  preserve_free : ∀ (a : 𝔸) (f : sig.free) {m : ℕ} (args : sig.freeArgs f → A m),
+    toFun (m + sig.freeDepth f) (A.free f ⟨a, args⟩) = B.free f ⟨a, toFun m ∘ args⟩
+  preserve_bound : ∀ (a : 𝔸) (f : sig.bound) {m : ℕ} (args : sig.boundArgs f → A m),
+    toFun (m + sig.boundDepth f) (A.bound f (⟪a⟫ args)) = B.bound f (⟪a⟫ toFun m ∘ args)
+
+instance {A B : Algebra 𝔸 sig} : CoeFun (Algebra.Hom 𝔸 sig A B) (fun _ => (n : ℕ) → (A n →ₑ[𝔸] B n)) where
+  coe h := h.toFun
+
+end AlgebraDefinition
+
+section Algebra
+
+variable {𝔸 : Type} [Infinite 𝔸] [Countable 𝔸] [DecidableEq 𝔸]
+variable {sig : Signature}
+
+namespace Algebra
+
+def Hom.id (A : Algebra 𝔸 sig) : Hom 𝔸 sig A A where
+  toFun _ := EquivariantMap.id 𝔸
+  preserve_pure _ _ _ := rfl
+  preserve_free _ _ _ _ := rfl
+  preserve_bound _ _ _ _ := rfl
+
+def Hom.comp {A B C : Algebra 𝔸 sig} (h : Hom 𝔸 sig B C) (g : Hom 𝔸 sig A B) : Hom 𝔸 sig A C where
+  toFun n := EquivariantMap.comp 𝔸 (h n) (g n)
+  preserve_pure f m args := by
+    rw [EquivariantMap.comp_app, g.preserve_pure, h.preserve_pure]
+    rfl
+  preserve_free a f m args := by
+    rw [EquivariantMap.comp_app, g.preserve_free, h.preserve_free]
+    rfl
+  preserve_bound a f m args := by
+    rw [EquivariantMap.comp_app, g.preserve_bound, h.preserve_bound]
+    rfl
+
+instance : Category (Algebra 𝔸 sig) where
+  Hom := Hom 𝔸 sig
+  id := Hom.id
+  comp h g := Hom.comp g h
+
+def forgetfulFunctor : Algebra 𝔸 sig ⥤ Nom 𝔸 where
+  obj X := X 0
+  map f := f.toFun 0
+
+end Algebra
+
+end Algebra
+
+end NominalAlgebra