diff --git a/core/src/main/scala/cats/Bitraverse.scala b/core/src/main/scala/cats/Bitraverse.scala
index a9a5f9c285..90804d364b 100644
--- a/core/src/main/scala/cats/Bitraverse.scala
+++ b/core/src/main/scala/cats/Bitraverse.scala
@@ -14,7 +14,7 @@ trait Bitraverse[F[_, _]] extends Bifoldable[F] with Bifunctor[F] { self =>
     bitraverse(fab)(identity, identity)
 
   /** If F and G are both [[cats.Bitraverse]] then so is their composition F[G[_, _], G[_, _]] */
-  def compose[G[_, _]](implicit ev: Bitraverse[G]): Bifoldable[Lambda[(A, B) => F[G[A, B], G[A, B]]]] =
+  def compose[G[_, _]](implicit ev: Bitraverse[G]): Bitraverse[Lambda[(A, B) => F[G[A, B], G[A, B]]]] =
     new CompositeBitraverse[F, G] {
       val F = self
       val G = ev
diff --git a/laws/src/main/scala/cats/laws/discipline/Eq.scala b/laws/src/main/scala/cats/laws/discipline/Eq.scala
index d79a1a6a84..ed5879bb22 100644
--- a/laws/src/main/scala/cats/laws/discipline/Eq.scala
+++ b/laws/src/main/scala/cats/laws/discipline/Eq.scala
@@ -3,6 +3,7 @@ package laws
 package discipline
 
 import algebra.Eq
+import cats.std.string._
 import org.scalacheck.Arbitrary
 
 object eq {
@@ -21,6 +22,12 @@ object eq {
     }
   }
 
+  /** Create an approximation of Eq[Show[A]] by using function1Eq[A, String] */
+  implicit def showEq[A: Arbitrary]: Eq[Show[A]] =
+    Eq.by[Show[A], A => String] { showInstance =>
+      (a: A) => showInstance.show(a)
+    }
+
   // Temporary, see https://github.com/non/algebra/pull/82
   implicit def tuple2Eq[A, B](implicit A: Eq[A], B: Eq[B]): Eq[(A, B)] =
     new Eq[(A, B)] {
diff --git a/tests/src/test/scala/cats/tests/BifoldableTests.scala b/tests/src/test/scala/cats/tests/BifoldableTests.scala
new file mode 100644
index 0000000000..2c121909bd
--- /dev/null
+++ b/tests/src/test/scala/cats/tests/BifoldableTests.scala
@@ -0,0 +1,15 @@
+package cats
+package tests
+
+import cats.data.Xor
+import cats.laws.discipline.{BifoldableTests, SerializableTests}
+import cats.laws.discipline.arbitrary._
+
+class BifoldableTest extends CatsSuite {
+  type EitherXor[A, B] = Either[Xor[A, B], Xor[A, B]]
+  val eitherComposeXor: Bifoldable[EitherXor] =
+    Bifoldable[Either].compose[Xor]
+
+  checkAll("Either compose Xor", BifoldableTests(eitherComposeXor).bifoldable[Int, Int, Int])
+  checkAll("Bifoldable[Either compose Xor]", SerializableTests.serializable(eitherComposeXor))
+}
diff --git a/tests/src/test/scala/cats/tests/BitraverseTests.scala b/tests/src/test/scala/cats/tests/BitraverseTests.scala
new file mode 100644
index 0000000000..e981925d80
--- /dev/null
+++ b/tests/src/test/scala/cats/tests/BitraverseTests.scala
@@ -0,0 +1,16 @@
+package cats
+package tests
+
+import cats.data.Xor
+import cats.laws.discipline.{BitraverseTests, SerializableTests}
+import cats.laws.discipline.arbitrary._
+import cats.laws.discipline.eq.tuple2Eq
+
+class BitraverseTest extends CatsSuite {
+  type XorTuple2[A, B] = Xor[(A, B), (A, B)]
+  val xorComposeTuple2: Bitraverse[XorTuple2] =
+    Bitraverse[Xor].compose[Tuple2]
+
+  checkAll("Xor compose Tuple2", BitraverseTests(xorComposeTuple2).bitraverse[Option, Int, Int, Int, String, String, String])
+  checkAll("Bitraverse[Xor compose Tuple2]", SerializableTests.serializable(xorComposeTuple2))
+}
diff --git a/tests/src/test/scala/cats/tests/CategoryTests.scala b/tests/src/test/scala/cats/tests/CategoryTests.scala
new file mode 100644
index 0000000000..b93cc732d6
--- /dev/null
+++ b/tests/src/test/scala/cats/tests/CategoryTests.scala
@@ -0,0 +1,19 @@
+package cats
+package tests
+
+import algebra.laws.GroupLaws
+
+import cats.arrow.Category
+import cats.laws.discipline.{MonoidKTests, SerializableTests}
+import cats.laws.discipline.eq.function1Eq
+
+class CategoryTest extends CatsSuite {
+  val functionCategory = Category[Function1]
+  type Endo[A] = Function1[A, A]
+
+  checkAll("Category[Function1].algebraK", MonoidKTests[Endo](functionCategory.algebraK).monoidK[Int])
+  checkAll("Category[Function1].algebraK", SerializableTests.serializable(functionCategory.algebraK))
+
+  val functionAlgebra = functionCategory.algebra[Int]
+  checkAll("Category[Function1].algebra[Int]", GroupLaws[Endo[Int]].monoid(functionAlgebra))
+}
diff --git a/tests/src/test/scala/cats/tests/ComposeTest.scala b/tests/src/test/scala/cats/tests/ComposeTest.scala
new file mode 100644
index 0000000000..74019444bb
--- /dev/null
+++ b/tests/src/test/scala/cats/tests/ComposeTest.scala
@@ -0,0 +1,19 @@
+package cats
+package tests
+
+import algebra.laws.GroupLaws
+
+import cats.arrow.Compose
+import cats.laws.discipline.{SemigroupKTests, SerializableTests}
+import cats.laws.discipline.eq.function1Eq
+
+class ComposeTest extends CatsSuite {
+  val functionCompose = Compose[Function1]
+  type Endo[A] = Function1[A, A]
+
+  checkAll("Compose[Function1].algebraK", SemigroupKTests[Endo](functionCompose.algebraK).semigroupK[Int])
+  checkAll("Compose[Function1].algebraK", SerializableTests.serializable(functionCompose.algebraK))
+
+  val functionAlgebra = functionCompose.algebra[Int]
+  checkAll("Compose[Function1].algebra[Int]", GroupLaws[Endo[Int]].semigroup(functionAlgebra))
+}
diff --git a/tests/src/test/scala/cats/tests/FunctorTests.scala b/tests/src/test/scala/cats/tests/FunctorTests.scala
new file mode 100644
index 0000000000..40ee82e96c
--- /dev/null
+++ b/tests/src/test/scala/cats/tests/FunctorTests.scala
@@ -0,0 +1,23 @@
+package cats
+package tests
+
+import cats.data.Xor
+import cats.functor.Contravariant
+import cats.laws.discipline.{ContravariantTests, FunctorTests, SerializableTests}
+import cats.laws.discipline.arbitrary._
+import cats.laws.discipline.eq.showEq
+
+class FunctorTest extends CatsSuite {
+  type OptionXor[A] = Option[Xor[String, A]]
+
+  val optionXorFunctor: Functor[OptionXor] =
+    Functor[Option].compose[Xor[String, ?]]
+  checkAll("Option compose Xor", FunctorTests(optionXorFunctor).functor[Int, Int, Int])
+  checkAll("Functor[Option compose Xor]", SerializableTests.serializable(optionXorFunctor))
+
+  type OptionShow[A] = Option[Show[A]]
+  val optionShowContravariant: Contravariant[OptionShow] =
+      Functor[Option].composeWithContravariant[Show]
+  checkAll("Option composeWithContravariant Show", ContravariantTests(optionShowContravariant).contravariant[Int, Int, Int])
+  checkAll("Contravariant[Option composeWithContravariant Show]", SerializableTests.serializable(optionShowContravariant))
+}
diff --git a/tests/src/test/scala/cats/tests/MonadStateTests.scala b/tests/src/test/scala/cats/tests/MonadStateTests.scala
new file mode 100644
index 0000000000..3c9d825b28
--- /dev/null
+++ b/tests/src/test/scala/cats/tests/MonadStateTests.scala
@@ -0,0 +1,22 @@
+package cats
+package tests
+
+import cats.data.State
+
+class MonadStateTest extends CatsSuite {
+  val testInstance = MonadState[State[Int, ?], Int]
+
+  test("MonadState#modify identity does not modify the state") {
+    forAll { (i: Int) =>
+      val x = testInstance.modify(identity).runS(i).value
+      x should === (i)
+    }
+  }
+
+  test("MonadState#inspect identity does not modify the state") {
+    forAll { (i: Int) =>
+      val x = testInstance.inspect(identity).runA(i).value
+      x should === (i)
+    }
+  }
+}