diff --git a/src/algebra/free.js b/src/algebra/free.js
index dd5d584..ca81457 100644
--- a/src/algebra/free.js
+++ b/src/algebra/free.js
@@ -1,7 +1,8 @@
 import { Algebra, Foldable, Monad, Traversable } from './interfaces.js'
 import { curry } from '../..//vendor/izuna/src/index.js'
+import { Reducer } from './utilities.js'
 
-/** @import { Applicative, ApplicativeTypeRef, Apply, Chain, Foldable as FoldableT, Functor, Monad as MonadT, Morphism, Traversable as TraversableT } from './types.js' */
+/** @import { Applicative, ApplicativeTypeRef, Apply, Chain, Comonad, Foldable as FoldableT, Functor, Monad as MonadT, Morphism, Traversable as TraversableT } from './types.js' */
 
 
 /** @import { InferredMorphism } from './types.js' */
@@ -10,17 +11,9 @@ const kleisli = curry((f, g, x) => f(x).chain(g))
 
 /**
  * @template T
- * @type {Pure<T> | Impure<T>} Free
+ * @typedef {Pure<T> | Impure<T>} Free
  */
-export class Free {
-  /**
-   * @template T
-   * @param {T} value 
-   */
-  static of(value) {
-    return new Pure(value)
-  }
-}
+const Free = {}
 
 /** @template T */
 export class Pure extends Algebra(Monad, Traversable) {
@@ -35,14 +28,14 @@ export class Pure extends Algebra(Monad, Traversable) {
   }
 
   /**
-   * @param {InferredMorphism<T>} f 
+   * @type {Chain<T>['chain']}
    */
   chain(f) {
     return f(this.#value)
   }
 
   /**
-   * @param {InferredMorphism<T>} f 
+   * @type {Functor<T>['map']}
    */
   map(f) {
     return pure(f(this.#value))
@@ -50,11 +43,13 @@ export class Pure extends Algebra(Monad, Traversable) {
 
   /**
    * @template U
+   * @type {Apply<T>['ap']}
    * @this {Pure<Morphism<U, T>>}
-   * @param {Free<T>} b 
+   * @param {Pure<U>} b 
+   * @returns {Free<T>}
    */
   ap(b) {
-    return b.map(this.#value)
+    return /** @type {Free<T>} */ (b.map(this.#value))
   }
 
   /**
@@ -105,7 +100,7 @@ export class Impure extends Algebra(Monad, Foldable, Traversable) {
    * @type {Chain<T>['chain']}
    * @template U
    * @param {(value: T) => Free<U>} f 
-   * @returns {Chain<T>}
+   * @returns {Free<T>}
    */
   chain(f) {
     return impure(this.#value, kleisli(this.#next, f))
@@ -114,33 +109,36 @@ export class Impure extends Algebra(Monad, Foldable, Traversable) {
   /**
    * @template {Functor<T>} U
    * @type {Functor<T>['map']}
-   * @param {(value: T) => U} f 
-   * @returns {Functor<T>}
+   * @param {Morphism<T, U>} f 
+   * @returns {Free<T>}
    */
   map(f) {
     return impure(
       this.#value,
-      y => f(y).map(f)
+      y => /** @type {Free<U>} */(f(y).map(f))
     )
   }
 
   /**
+   * @template U
    * @type {Apply<T>['ap']}
+   * @this {Free<Morphism<U, T>>}
+   * @param {Free<U>} b 
+   * @returns {Free<T>}
    */
   ap(b) {
-    return this.chain(f => b.map(f))
+    return /** @type {Free<T>} */ (this.chain(f =>
+      /** @type {Free<T>} */(b.map(f))
+    ))
   }
 
   /**
    * @type {FoldableT<T>['reduce']}
    */
   reduce(f, acc) {
-    const fb = f(acc, this.#value)
-    const rest = this.#next(this.#value).reduce(f, acc)
-
-    return rest.ap(
-      fb.map(b => next => impure(b, () => next))
-    )
+    const Const = Reducer(f, acc)
+    // @ts-ignore
+    return this.traverse(Const, x => new Const(x)).value
   }
 
   /**
@@ -154,12 +152,20 @@ export class Impure extends Algebra(Monad, Foldable, Traversable) {
    * @returns {Applicative<Free<U>>}
    */
   traverse(A, f) {
-    const fb = f(this.#value)
-    const rest = this.#next(this.#value).traverse(A, f)
+    const next = this.#next(this.#value)
 
-    return rest.ap(
-      fb.map(b => next => impure(b, () => next))
-    )
+    if (next instanceof Pure) {
+      return next.traverse(A, f)
+    } else {
+      const fb = f(this.#value)
+      const rest = next.traverse(A, f)
+
+      return /** @type {Applicative<Free<U>>} */ (rest.ap(
+        fb.map(b => next => impure(b, () =>
+          /** @type {Free<U>} */(next)
+        ))
+      ))
+    }
   }
 
   run() {
@@ -192,6 +198,6 @@ export const liftF = x => impure(x, pure)
 import { id, Identity } from './identity.js'
 import { Option, some } from './option.js'
 
-const a = liftF(1).chain(() => liftF(2)).traverse(Option, x => some(1))
-console.log(a.chain(x => x).run())
+const a = liftF(1).chain(() => liftF(2)).reduce((acc, x) => acc + x, 0)
+console.log(a)
 
diff --git a/src/algebra/identity.js b/src/algebra/identity.js
index d892b41..efba1b3 100644
--- a/src/algebra/identity.js
+++ b/src/algebra/identity.js
@@ -1,6 +1,6 @@
 import { Algebra, Comonad, Monad } from './interfaces.js'
 
-/** @import { Morphism } from './types.js' */
+/** @import { Apply, Morphism } from './types.js' */
 
 /** @template T */
 export class Identity extends Algebra(Monad, Comonad) {
@@ -28,17 +28,18 @@ export class Identity extends Algebra(Monad, Comonad) {
    * @param {Morphism<T, U>} f 
    */
   map(f) {
+    console.log(this.toString(), f.toString())
     return id(f(this.#value))
   }
 
   /**
    * @template U
-   * @this {Identity<Morphism<U, T>>}
-   * @param {Identity<U>} b 
-   * @returns {Identity<T>}
+   * @type {Apply<T>['ap']}
+   * @param {Apply<Morphism<T, U>>} b 
+   * @returns {Apply<U>}
    */
   ap(b) {
-    return b.map(this.#value)
+    return /** @type {Apply<U>} */ (b.map(f => f(this.#value)))
   }
 
   /**
@@ -59,6 +60,10 @@ export class Identity extends Algebra(Monad, Comonad) {
   extract() {
     return this.#value
   }
+
+  toString() {
+    return `Identity(${this.#value})`
+  }
 }
 
 export const id = value => new Identity(value)
diff --git a/src/algebra/utilities.js b/src/algebra/utilities.js
new file mode 100644
index 0000000..88cdb12
--- /dev/null
+++ b/src/algebra/utilities.js
@@ -0,0 +1,26 @@
+/**
+ * @template T, U
+ * @param {(acc: U, value: T) => U} f
+ * @param {U} acc
+ */
+export const Reducer = (f, acc) => {
+  /** @template A */
+  return class Const {
+    /** @param {A} value */
+    constructor(value) {
+      this.value = value
+    }
+
+    static of() { return new Const(acc) }
+
+    map() { return this }
+
+    /**
+     * @this {Const<T>}
+     * @param {Const<U>} b
+     */
+    ap(b) {
+      return new Const(f(b.value, this.value))
+    }
+  }
+}