Add type aliases for composed type constructors

core/src/main/scala-3/cats/derived/Derived.scala

@@ -12,15 +12,18 @@ object Derived:
   extension [A](derived: Derived[A]) def instance: A = derived
   given [A]: Conversion[A, Derived[A]] = apply
 
-  type Or0[F[_]] = [x] =>> Or[F[x]]
-  type Or1[F[_[_]]] = [x[_]] =>> Or[F[x]]
-  type Or11[F[_[_[_]]]] = [x[_[_]]] =>> Or[F[x]]
-  type Or2[F[_[_, _]]] = [x[_, _]] =>> Or[F[x]]
+  infix type >>>[F[_], G[_]] = [x] =>> G[F[x]]
+  infix type <<<[F[_], G[_]] = [x] =>> F[G[x]]
+
+  type Or0[F[_]] = [x] =>> Derived.Or[F[x]]
+  type Or1[F[_[_]]] = [x[_]] =>> Derived.Or[F[x]]
+  type Or11[F[_[_[_]]]] = [x[_[_]]] =>> Derived.Or[F[x]]
+  type Or2[F[_[_, _]]] = [x[_, _]] =>> Derived.Or[F[x]]
 
   opaque type Or[A] = A
   object Or extends OrInstances:
     def apply[A](instance: A): Or[A] = instance
-    extension [A](or: Or[A]) def unify: A = or
+    extension [A](derived: Or[A]) def unify: A = derived
     extension [I[f[_], t] <: K0.Instances[f, t], F[_], T](inst: I[Or0[F], T])
       @targetName("unifyK0") def unify: I[F, T] = inst
     extension [I[f[_[_]], t[_]] <: K1.Instances[f, t], F[_[_]], T[_]](inst: I[Or1[F], T])

core/src/main/scala-3/cats/derived/DerivedAlternative.scala

@@ -1,6 +1,7 @@
 package cats.derived
 
 import cats.Alternative
+import cats.derived.Derived.<<<
 import shapeless3.deriving.K1.*
 
 import scala.annotation.*
@@ -25,10 +26,10 @@ object DerivedAlternative:
     summonInline[DerivedAlternative[F]].instance
 
   given nested[F[_], G[_]](using
-      F: => Or[F],
+      F: => DerivedAlternative.Or[F],
       G: => DerivedApplicative.Or[G]
-  ): DerivedAlternative[[x] =>> F[G[x]]] =
-    new Derived.Lazy(() => F.unify.compose(using G.unify)) with Alternative[[x] =>> F[G[x]]]:
+  ): DerivedAlternative[F <<< G] =
+    new Derived.Lazy(() => F.unify.compose(using G.unify)) with Alternative[F <<< G]:
       export delegate.*
 
   given product[F[_]](using inst: => ProductInstances[Or, F]): DerivedAlternative[F] =

core/src/main/scala-3/cats/derived/DerivedApplicative.scala

@@ -1,5 +1,6 @@
 package cats.derived
 
+import cats.derived.Derived.<<<
 import cats.{Applicative, Monoid}
 import shapeless3.deriving.Const
 import shapeless3.deriving.K1.*
@@ -30,15 +31,19 @@ object DerivedApplicative:
     def pure[A](x: A): T = T.empty
     def ap[A, B](ff: T)(fa: T): T = T.combine(ff, fa)
 
-  given nested[F[_], G[_]](using F: => Or[F], G: => Or[G]): DerivedApplicative[[x] =>> F[G[x]]] =
-    new Derived.Lazy(() => F.unify.compose(using G.unify)) with Applicative[[x] =>> F[G[x]]]:
+  given nested[F[_], G[_]](using
+      F: => DerivedApplicative.Or[F],
+      G: => DerivedApplicative.Or[G]
+  ): DerivedApplicative[F <<< G] =
+    new Derived.Lazy(() => F.unify.compose(using G.unify)) with Applicative[F <<< G]:
       export delegate.*
 
   given [F[_]](using inst: => ProductInstances[Or, F]): DerivedApplicative[F] =
     Strict.product(using inst.unify)
 
   @deprecated("Kept for binary compatibility", "3.2.0")
-  protected given [F[_]: Or, G[_]: Or]: DerivedApplicative[[x] =>> F[G[x]]] = nested
+  protected given [F[_]: DerivedApplicative.Or, G[_]: DerivedApplicative.Or]: DerivedApplicative[[x] =>> F[G[x]]] =
+    nested
 
   trait Product[T[f[_]] <: Applicative[f], F[_]](using inst: ProductInstances[T, F])
       extends Applicative[F],

core/src/main/scala-3/cats/derived/DerivedApply.scala

@@ -1,5 +1,6 @@
 package cats.derived
 
+import cats.derived.Derived.<<<
 import cats.{Apply, Semigroup}
 import shapeless3.deriving.Const
 import shapeless3.deriving.K1.*
@@ -30,18 +31,18 @@ object DerivedApply:
     def ap[A, B](ff: T)(fa: T): T = T.combine(ff, fa)
     def map[A, B](fa: T)(f: A => B): T = fa
 
-  given nested[F[_], G[_]](using F: => Or[F], G: => Or[G]): DerivedApply[[x] =>> F[G[x]]] =
-    new Derived.Lazy(() => F.unify.compose(using G.unify)) with Apply[[x] =>> F[G[x]]]:
+  given nested[F[_], G[_]](using F: => DerivedApply.Or[F], G: => DerivedApply.Or[G]): DerivedApply[F <<< G] =
+    new Derived.Lazy(() => F.unify.compose(using G.unify)) with Apply[F <<< G]:
       export delegate.*
 
   given [F[_]](using inst: => ProductInstances[Or, F]): DerivedApply[F] =
     Strict.product(using inst.unify)
 
   @deprecated("Kept for binary compatibility", "3.2.0")
-  protected given [F[_]: Or, G[_]: Or]: DerivedApply[[x] =>> F[G[x]]] = nested
+  protected given [F[_]: DerivedApply.Or, G[_]: DerivedApply.Or]: DerivedApply[[x] =>> F[G[x]]] = nested
 
   trait Product[T[f[_]] <: Apply[f], F[_]](using inst: ProductInstances[T, F]) extends Apply[F]:
-    private lazy val F = new DerivedFunctor.Generic[T, F] {}
+    private lazy val F = DerivedFunctor.Strict.product(using inst.widen).instance
     final override def map[A, B](fa: F[A])(f: A => B): F[B] = F.map(fa)(f)
     final override def ap[A, B](ff: F[A => B])(fa: F[A]): F[B] =
       inst.map2(ff, fa)([f[_]] => (F: T[f], ff: f[A => B], fa: f[A]) => F.ap(ff)(fa))

core/src/main/scala-3/cats/derived/DerivedContravariant.scala

@@ -1,6 +1,7 @@
 package cats.derived
 
 import cats.Contravariant
+import cats.derived.Derived.<<<
 import shapeless3.deriving.Const
 import shapeless3.deriving.K1.*
 
@@ -30,15 +31,19 @@ object DerivedContravariant:
   given [T]: DerivedContravariant[Const[T]] = new Contravariant[Const[T]]:
     def contramap[A, B](fa: T)(f: B => A): T = fa
 
-  given nested[F[_], G[_]](using F: DerivedFunctor.Or[F], G: => Or[G]): DerivedContravariant[[x] =>> F[G[x]]] =
-    new Derived.Lazy(() => F.unify.composeContravariant(using G.unify)) with Contravariant[[x] =>> F[G[x]]]:
+  given nested[F[_], G[_]](using
+      F: DerivedFunctor.Or[F],
+      G: => DerivedContravariant.Or[G]
+  ): DerivedContravariant[F <<< G] =
+    new Derived.Lazy(() => F.unify.composeContravariant(using G.unify)) with Contravariant[F <<< G]:
       export delegate.*
 
   given [F[_]](using inst: => Instances[Or, F]): DerivedContravariant[F] =
     generic(using inst.unify)
 
   @deprecated("Kept for binary compatibility", "3.2.0")
-  protected given [F[_]: DerivedFunctor.Or, G[_]: Or]: DerivedContravariant[[x] =>> F[G[x]]] = nested
+  protected given [F[_]: DerivedFunctor.Or, G[_]: DerivedContravariant.Or]: DerivedContravariant[[x] =>> F[G[x]]] =
+    nested
 
   private def generic[F[_]: InstancesOf[Contravariant]]: DerivedContravariant[F] =
     new Generic[Contravariant, F] {}

core/src/main/scala-3/cats/derived/DerivedEmpty.scala

@@ -24,12 +24,15 @@ object DerivedEmpty:
     import Strict.given
     summonInline[DerivedEmpty[A]].instance
 
-  given product[A: ProductInstancesOf[Or]]: DerivedEmpty[A] = Strict.product(using ProductInstances.unify)
-  inline given coproduct[A: CoproductGeneric]: DerivedEmpty[A] = Strict.coproduct
+  given product[A: ProductInstancesOf[DerivedEmpty.Or]]: DerivedEmpty[A] =
+    Strict.product(using ProductInstances.unify)
+
+  inline given coproduct[A: CoproductGeneric]: DerivedEmpty[A] =
+    Strict.coproduct
 
   object Strict:
     given product[A: ProductInstancesOf[Empty]]: DerivedEmpty[A] =
       Empty(ProductInstances.construct([a] => (A: Empty[a]) => A.empty))
 
     inline given coproduct[A: CoproductGeneric]: DerivedEmpty[A] =
-      Empty(CoproductGeneric.withOnly[Or, A]([a <: A] => (A: Or[a]) => A.unify.empty))
+      Empty(CoproductGeneric.withOnly[DerivedEmpty.Or, A]([a <: A] => (A: DerivedEmpty.Or[a]) => A.unify.empty))

core/src/main/scala-3/cats/derived/DerivedEmptyK.scala

@@ -1,6 +1,7 @@
 package cats.derived
 
 import alleycats.{Empty, EmptyK}
+import cats.derived.Derived.<<<
 import shapeless3.deriving.Const
 import shapeless3.deriving.K1.*
 
@@ -32,35 +33,39 @@ object DerivedEmptyK:
   given [T](using T: Empty[T]): DerivedEmptyK[Const[T]] = new EmptyK[Const[T]]:
     def empty[A]: T = T.empty
 
-  given nested[F[_], G[_]](using F: => Or[F]): DerivedEmptyK[[x] =>> F[G[x]]] =
-    new EmptyK[[x] =>> F[G[x]]]:
+  given nested[F[_], G[_]](using F: => DerivedEmptyK.Or[F]): DerivedEmptyK[F <<< G] =
+    new EmptyK[F <<< G]:
       lazy val f = F.unify
       def empty[A]: F[G[A]] = f.empty
 
   given nested[F[_], G[_]](using
-      NotGiven[Or[F]]
-  )(using F: DerivedPure.Or[F], G: => Or[G]): DerivedEmptyK[[x] =>> F[G[x]]] =
-    new EmptyK[[x] =>> F[G[x]]]:
+      NotGiven[DerivedEmptyK.Or[F]]
+  )(using F: DerivedPure.Or[F], G: => DerivedEmptyK.Or[G]): DerivedEmptyK[F <<< G] =
+    new EmptyK[F <<< G]:
       val f = F.unify
       lazy val g = G.unify
       def empty[A]: F[G[A]] = f.pure(g.empty)
 
-  given product[F[_]: ProductInstancesOf[Or]]: DerivedEmptyK[F] = Strict.product(using ProductInstances.unify)
-  inline given coproduct[F[_]: CoproductGeneric]: DerivedEmptyK[F] = Strict.coproduct
+  given product[F[_]: ProductInstancesOf[DerivedEmptyK.Or]]: DerivedEmptyK[F] =
+    Strict.product(using ProductInstances.unify)
+
+  inline given coproduct[F[_]: CoproductGeneric]: DerivedEmptyK[F] =
+    Strict.coproduct
 
   @deprecated("Kept for binary compatibility", "3.2.0")
-  protected given [F[_], G[_]](using F: Or[F]): DerivedEmptyK[[x] =>> F[G[x]]] =
+  protected given [F[_], G[_]](using F: DerivedEmptyK.Or[F]): DerivedEmptyK[[x] =>> F[G[x]]] =
     nested(using F)
 
   @deprecated("Kept for binary compatibility", "3.2.0")
   protected given [F[_], G[_]](using
-      ev: NotGiven[Or[F]]
-  )(using DerivedPure.Or[F], Or[G]): DerivedEmptyK[[x] =>> F[G[x]]] =
+      ev: NotGiven[DerivedEmptyK.Or[F]]
+  )(using DerivedPure.Or[F], DerivedEmptyK.Or[G]): DerivedEmptyK[[x] =>> F[G[x]]] =
     nested(using ev)
 
   object Strict:
     given product[F[_]: ProductInstancesOf[EmptyK]]: DerivedEmptyK[F] = new EmptyK[F]:
       def empty[A]: F[A] = ProductInstances.construct([f[_]] => (F: EmptyK[f]) => F.empty[A])
 
     inline given coproduct[F[_]: CoproductGeneric]: DerivedEmptyK[F] =
-      CoproductGeneric.withOnly[Or, EmptyK[F]]([f[x] <: F[x]] => (F: Or[f]) => F.unify.asInstanceOf[EmptyK[F]])
+      CoproductGeneric.withOnly[DerivedEmptyK.Or, EmptyK[F]]:
+        [f[x] <: F[x]] => (F: DerivedEmptyK.Or[f]) => F.unify.asInstanceOf[EmptyK[F]]

core/src/main/scala-3/cats/derived/DerivedFoldable.scala

@@ -1,8 +1,9 @@
 package cats.derived
 
+import cats.derived.Derived.<<<
 import cats.{Eval, Foldable}
-import shapeless3.deriving.{Const, Continue}
 import shapeless3.deriving.K1.*
+import shapeless3.deriving.{Const, Continue}
 
 import scala.annotation.*
 import scala.compiletime.*
@@ -31,18 +32,18 @@ object DerivedFoldable:
     def foldLeft[A, B](fa: T, b: B)(f: (B, A) => B): B = b
     def foldRight[A, B](fa: T, lb: Eval[B])(f: (A, Eval[B]) => Eval[B]): Eval[B] = lb
 
-  given nested[F[_], G[_]](using F: => Or[F], G: => Or[G]): DerivedFoldable[[x] =>> F[G[x]]] =
-    new Derived.Lazy(() => F.unify.compose(using G.unify)) with Foldable[[x] =>> F[G[x]]]:
+  given nested[F[_], G[_]](using F: => DerivedFoldable.Or[F], G: => DerivedFoldable.Or[G]): DerivedFoldable[F <<< G] =
+    new Derived.Lazy(() => F.unify.compose(using G.unify)) with Foldable[F <<< G]:
       export delegate.*
 
-  given [F[_]: ProductInstancesOf[Or]]: DerivedFoldable[F] =
+  given [F[_]: ProductInstancesOf[DerivedFoldable.Or]]: DerivedFoldable[F] =
     Strict.product(using ProductInstances.unify)
 
   given [F[_]](using => CoproductInstances[Or, F]): DerivedFoldable[F] =
     Strict.coproduct
 
   @deprecated("Kept for binary compatibility", "3.2.0")
-  protected given [F[_]: Or, G[_]: Or]: DerivedFoldable[[x] =>> F[G[x]]] = nested
+  protected given [F[_]: DerivedFoldable.Or, G[_]: DerivedFoldable.Or]: DerivedFoldable[[x] =>> F[G[x]]] = nested
 
   trait Product[T[f[_]] <: Foldable[f], F[_]](using inst: ProductInstances[T, F]) extends Foldable[F]:
     final override def foldLeft[A, B](fa: F[A], b: B)(f: (B, A) => B): B =

core/src/main/scala-3/cats/derived/DerivedFunctor.scala

@@ -1,6 +1,7 @@
 package cats.derived
 
 import cats.Functor
+import cats.derived.Derived.<<<
 import shapeless3.deriving.Const
 import shapeless3.deriving.K1.*
 
@@ -31,21 +32,24 @@ object DerivedFunctor:
   given [T]: DerivedFunctor[Const[T]] = new Functor[Const[T]]:
     def map[A, B](fa: T)(f: A => B): T = fa
 
-  given nested[F[_], G[_]](using F: => Or[F], G: => Or[G]): DerivedFunctor[[x] =>> F[G[x]]] =
-    new Derived.Lazy(() => F.unify.compose(using G.unify)) with Functor[[x] =>> F[G[x]]]:
+  given nested[F[_], G[_]](using F: => DerivedFunctor.Or[F], G: => DerivedFunctor.Or[G]): DerivedFunctor[F <<< G] =
+    new Derived.Lazy(() => F.unify.compose(using G.unify)) with Functor[F <<< G]:
       export delegate.*
 
   given nested[F[_], G[_]](using
       F: DerivedContravariant.Or[F],
       G: DerivedContravariant.Or[G]
-  ): DerivedFunctor[[x] =>> F[G[x]]] =
+  ): DerivedFunctor[F <<< G] =
     F.unify.compose(using G.unify)
 
   given [F[_]](using inst: => Instances[Or, F]): DerivedFunctor[F] =
     generic(using inst.unify)
 
   @deprecated("Kept for binary compatibility", "3.2.0")
-  protected given [F[_], G[_]](using F: Or[F], G: Or[G]): DerivedFunctor[[x] =>> F[G[x]]] =
+  protected given [F[_], G[_]](using
+      F: DerivedFunctor.Or[F],
+      G: DerivedFunctor.Or[G]
+  ): DerivedFunctor[[x] =>> F[G[x]]] =
     nested(using F, G)
 
   @deprecated("Kept for binary compatibility", "3.2.0")

core/src/main/scala-3/cats/derived/DerivedInvariant.scala

@@ -1,6 +1,7 @@
 package cats.derived
 
 import cats.Invariant
+import cats.derived.Derived.<<<
 import shapeless3.deriving.Const
 import shapeless3.deriving.K1.*
 
@@ -30,15 +31,18 @@ object DerivedInvariant:
   given [T]: DerivedInvariant[Const[T]] = new Invariant[Const[T]]:
     def imap[A, B](fa: T)(f: A => B)(g: B => A): T = fa
 
-  given nested[F[_], G[_]](using F: => Or[F], G: => Or[G]): DerivedInvariant[[x] =>> F[G[x]]] =
-    new Derived.Lazy(() => F.unify.compose(using G.unify)) with Invariant[[x] =>> F[G[x]]]:
+  given nested[F[_], G[_]](using
+      F: => DerivedInvariant.Or[F],
+      G: => DerivedInvariant.Or[G]
+  ): DerivedInvariant[F <<< G] =
+    new Derived.Lazy(() => F.unify.compose(using G.unify)) with Invariant[F <<< G]:
       export delegate.*
 
   given [F[_]](using inst: => Instances[Or, F]): DerivedInvariant[F] =
     generic(using inst.unify)
 
   @deprecated("Kept for binary compatibility", "3.2.0")
-  protected given [F[_]: Or, G[_]: Or]: DerivedInvariant[[x] =>> F[G[x]]] = nested
+  protected given [F[_]: DerivedInvariant.Or, G[_]: DerivedInvariant.Or]: DerivedInvariant[[x] =>> F[G[x]]] = nested
 
   private def generic[F[_]: InstancesOf[Invariant]]: DerivedInvariant[F] =
     new Generic[Invariant, F] {}

core/src/main/scala-3/cats/derived/DerivedMonoidK.scala

@@ -1,5 +1,6 @@
 package cats.derived
 
+import cats.derived.Derived.<<<
 import cats.{Monoid, MonoidK}
 import shapeless3.deriving.Const
 import shapeless3.deriving.K1.*
@@ -32,14 +33,14 @@ object DerivedMonoidK:
     def empty[A]: T = T.empty
     def combineK[A](x: T, y: T): T = T.combine(x, y)
 
-  given nested[F[_], G[_]](using F: => Or[F]): DerivedMonoidK[[x] =>> F[G[x]]] =
-    new Derived.Lazy(() => F.unify.compose[G]) with MonoidK[[x] =>> F[G[x]]]:
+  given nested[F[_], G[_]](using F: => DerivedMonoidK.Or[F]): DerivedMonoidK[F <<< G] =
+    new Derived.Lazy(() => F.unify.compose[G]) with MonoidK[F <<< G]:
       export delegate.*
 
   given nested[F[_], G[_]](using
-      NotGiven[Or[F]]
-  )(using F: DerivedApplicative.Or[F], G: => Or[G]): DerivedMonoidK[[x] =>> F[G[x]]] =
-    new MonoidK[[x] =>> F[G[x]]]:
+      NotGiven[DerivedMonoidK.Or[F]]
+  )(using F: DerivedApplicative.Or[F], G: => DerivedMonoidK.Or[G]): DerivedMonoidK[F <<< G] =
+    new MonoidK[F <<< G]:
       val f = F.unify
       lazy val g = G.unify
       def empty[A]: F[G[A]] = f.pure(g.empty[A])
@@ -49,13 +50,13 @@ object DerivedMonoidK:
     Strict.product(using inst.unify)
 
   @deprecated("Kept for binary compatibility", "3.2.0")
-  protected given [F[_], G[_]](using F: Or[F]): DerivedMonoidK[[x] =>> F[G[x]]] =
+  protected given [F[_], G[_]](using F: DerivedMonoidK.Or[F]): DerivedMonoidK[[x] =>> F[G[x]]] =
     nested(using F)
 
   @deprecated("Kept for binary compatibility", "3.2.0")
   protected given [F[_], G[_]](using
-      ev: NotGiven[Or[F]]
-  )(using F: DerivedApplicative.Or[F], G: Or[G]): DerivedMonoidK[[x] =>> F[G[x]]] =
+      ev: NotGiven[DerivedMonoidK.Or[F]]
+  )(using F: DerivedApplicative.Or[F], G: DerivedMonoidK.Or[G]): DerivedMonoidK[[x] =>> F[G[x]]] =
     nested(using ev)
 
   trait Product[T[f[_]] <: MonoidK[f], F[_]](using inst: ProductInstances[T, F])

core/src/main/scala-3/cats/derived/DerivedNonEmptyAlternative.scala

@@ -1,6 +1,7 @@
 package cats.derived
 
 import cats.NonEmptyAlternative
+import cats.derived.Derived.<<<
 import shapeless3.deriving.K1.*
 
 import scala.annotation.*
@@ -25,10 +26,10 @@ object DerivedNonEmptyAlternative:
     summonInline[DerivedNonEmptyAlternative[F]].instance
 
   given nested[F[_], G[_]](using
-      F: => Or[F],
+      F: => DerivedNonEmptyAlternative.Or[F],
       G: => DerivedApplicative.Or[G]
-  ): DerivedNonEmptyAlternative[[x] =>> F[G[x]]] =
-    new Derived.Lazy(() => F.unify.compose(using G.unify)) with NonEmptyAlternative[[x] =>> F[G[x]]]:
+  ): DerivedNonEmptyAlternative[F <<< G] =
+    new Derived.Lazy(() => F.unify.compose(using G.unify)) with NonEmptyAlternative[F <<< G]:
       export delegate.*
 
   given product[F[_]](using inst: => ProductInstances[Or, F]): DerivedNonEmptyAlternative[F] =