module Pandora.Paradigm.Schemes.TUT where

import Pandora.Core.Functor (type (:.), type (:=), type (~>))
import Pandora.Pattern.Semigroupoid ((.))
import Pandora.Pattern.Category (identity, ($))
import Pandora.Pattern.Functor.Covariant (Covariant, Covariant ((-<$>-)), (-<$$>-), (-<$$$>-))
import Pandora.Pattern.Functor.Contravariant (Contravariant)
import Pandora.Pattern.Functor.Semimonoidal (Semimonoidal (multiply))
import Pandora.Pattern.Functor.Monoidal (Monoidal (unit))
import Pandora.Pattern.Functor.Extendable (Extendable ((<<=)))
import Pandora.Pattern.Functor.Distributive (Distributive ((-<<)))
import Pandora.Pattern.Functor.Bindable (Bindable ((=<<)))
import Pandora.Pattern.Functor.Bivariant ((<->))
import Pandora.Pattern.Functor.Adjoint (Adjoint ((-|), (|-)))
import Pandora.Pattern.Transformer.Liftable (Liftable (lift))
import Pandora.Pattern.Transformer.Lowerable (Lowerable (lower))
import Pandora.Paradigm.Primary.Algebraic.Exponential (type (<--))
import Pandora.Paradigm.Primary.Algebraic.Product ((:*:)((:*:)))
import Pandora.Paradigm.Primary.Algebraic.One (One (One))
import Pandora.Paradigm.Primary.Algebraic (point, extract)
import Pandora.Paradigm.Primary.Transformer.Flip (Flip (Flip))
import Pandora.Paradigm.Controlflow.Effect.Interpreted (Interpreted (Primary, run, unite))

newtype TUT ct ct' cu t t' u a = TUT (t :. u :. t' := a)

infix 3 <:<.>:>, >:<.>:>, <:<.>:<, >:<.>:<, <:>.<:>, >:>.<:>, <:>.<:<, >:>.<:<

type (<:<.>:>) = TUT Covariant Covariant Covariant
type (>:<.>:>) = TUT Contravariant Covariant Covariant
type (<:<.>:<) = TUT Covariant Covariant Contravariant
type (>:<.>:<) = TUT Contravariant Covariant Contravariant
type (<:>.<:>) = TUT Covariant Contravariant Covariant
type (>:>.<:>) = TUT Contravariant Contravariant Covariant
type (<:>.<:<) = TUT Covariant Contravariant Contravariant
type (>:>.<:<) = TUT Contravariant Contravariant Contravariant

instance Interpreted (TUT ct ct' cu t t' u) where
	type Primary (TUT ct ct' cu t t' u) a = t :. u :. t' := a
	run :: TUT ct ct' cu t t' u a -> Primary (TUT ct ct' cu t t' u) a
run ~(TUT (t :. (u :. t')) := a
x) = (t :. (u :. t')) := a
Primary (TUT ct ct' cu t t' u) a
x
	unite :: Primary (TUT ct ct' cu t t' u) a -> TUT ct ct' cu t t' u a
unite = Primary (TUT ct ct' cu t t' u) a -> TUT ct ct' cu t t' u a
forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *)
       (t' :: k -> k) (u :: k -> k) (a :: k).
((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a
TUT

instance (Covariant (->) (->) t, Covariant (->) (->) t', Covariant (->) (->) u) => Covariant (->) (->) (t <:<.>:> t' := u) where
	a -> b
f -<$>- :: (a -> b) -> (:=) (t <:<.>:> t') u a -> (:=) (t <:<.>:> t') u b
-<$>- TUT (t :. (u :. t')) := a
x = ((t :. (u :. t')) := b) -> (:=) (t <:<.>:> t') u b
forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *)
       (t' :: k -> k) (u :: k -> k) (a :: k).
((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a
TUT (((t :. (u :. t')) := b) -> (:=) (t <:<.>:> t') u b)
-> ((t :. (u :. t')) := b) -> (:=) (t <:<.>:> t') u b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ a -> b
f (a -> b) -> ((t :. (u :. t')) := a) -> (t :. (u :. t')) := b
forall (t :: * -> *) (u :: * -> *) (v :: * -> *)
       (category :: * -> * -> *) a b.
(Covariant category category t, Covariant category category u,
 Covariant category category v) =>
category a b -> category (t (u (v a))) (t (u (v b)))
-<$$$>- (t :. (u :. t')) := a
x

instance (Covariant (->) (->) t, Covariant (->) (->) t', Covariant (->) (->) u, Semimonoidal (->) (:*:) (:*:) t, Semimonoidal (->) (:*:) (:*:) u, Semimonoidal (->) (:*:) (:*:) t') => Semimonoidal (->) (:*:) (:*:) (t <:<.>:> t' := u) where
	multiply :: ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b)
-> (:=) (t <:<.>:> t') u (a :*: b)
multiply (TUT (t :. (u :. t')) := a
x :*: TUT (t :. (u :. t')) := b
y) = ((t :. (u :. t')) := (a :*: b)) -> (:=) (t <:<.>:> t') u (a :*: b)
forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *)
       (t' :: k -> k) (u :: k -> k) (a :: k).
((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a
TUT (((t :. (u :. t')) := (a :*: b))
 -> (:=) (t <:<.>:> t') u (a :*: b))
-> ((t :. (u :. t')) := (a :*: b))
-> (:=) (t <:<.>:> t') u (a :*: b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ forall k (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k).
Semimonoidal p source target t =>
p (source (t a) (t b)) (t (target a b))
forall (target :: * -> * -> *) (t :: * -> *) a b.
Semimonoidal (->) (:*:) target t =>
(t a :*: t b) -> t (target a b)
multiply @(->) @(:*:) ((t' a :*: t' b) -> t' (a :*: b))
-> ((u (t' a) :*: u (t' b)) -> u (t' a :*: t' b))
-> (u (t' a) :*: u (t' b))
-> u (t' (a :*: b))
forall (t :: * -> *) (u :: * -> *) (category :: * -> * -> *) a b.
(Covariant category category u, Covariant category category t) =>
category a b -> category (t (u a)) (t (u b))
-<$$>- forall k (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k).
Semimonoidal p source target t =>
p (source (t a) (t b)) (t (target a b))
forall (target :: * -> * -> *) (t :: * -> *) a b.
Semimonoidal (->) (:*:) target t =>
(t a :*: t b) -> t (target a b)
multiply @(->) @(:*:) ((u (t' a) :*: u (t' b)) -> u (t' (a :*: b)))
-> t (u (t' a) :*: u (t' b)) -> (t :. (u :. t')) := (a :*: b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>- (((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b))
-> t (u (t' a) :*: u (t' b))
forall k (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k).
Semimonoidal p source target t =>
p (source (t a) (t b)) (t (target a b))
multiply ((t :. (u :. t')) := a
x ((t :. (u :. t')) := a)
-> ((t :. (u :. t')) := b)
-> ((t :. (u :. t')) := a) :*: ((t :. (u :. t')) := b)
forall s a. s -> a -> s :*: a
:*: (t :. (u :. t')) := b
y)

instance (Covariant (->) (->) t, Semimonoidal (<--) (:*:) (:*:) t, Covariant (->) (->) u, Semimonoidal (<--) (:*:) (:*:) u, Covariant (->) (->) t', Semimonoidal (<--) (:*:) (:*:) t') => Semimonoidal (<--) (:*:) (:*:) (t <:<.>:> t' := u) where
	multiply :: ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b)
<-- (:=) (t <:<.>:> t') u (a :*: b)
multiply = ((:=) (t <:<.>:> t') u (a :*: b)
 -> (:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b)
-> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b)
   <-- (:=) (t <:<.>:> t') u (a :*: b)
forall (v :: * -> * -> *) a e. v e a -> Flip v a e
Flip (((:=) (t <:<.>:> t') u (a :*: b)
  -> (:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b)
 -> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b)
    <-- (:=) (t <:<.>:> t') u (a :*: b))
-> ((:=) (t <:<.>:> t') u (a :*: b)
    -> (:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b)
-> ((:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b)
   <-- (:=) (t <:<.>:> t') u (a :*: b)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ \(TUT (t :. (u :. t')) := (a :*: b)
xys) ->
		let Flip t (a :*: b) -> t a :*: t b
f = forall k (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k).
Semimonoidal p source target t =>
p (source (t a) (t b)) (t (target a b))
forall (t :: * -> *) a b.
Semimonoidal (<--) (:*:) (:*:) t =>
(t a :*: t b) <-- t (a :*: b)
multiply @(<--) @(:*:) @(:*:) in
		let Flip u (a :*: b) -> u a :*: u b
g = forall k (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k).
Semimonoidal p source target t =>
p (source (t a) (t b)) (t (target a b))
forall (t :: * -> *) a b.
Semimonoidal (<--) (:*:) (:*:) t =>
(t a :*: t b) <-- t (a :*: b)
multiply @(<--) @(:*:) @(:*:) in
		let Flip t' (a :*: b) -> t' a :*: t' b
h = forall k (p :: * -> * -> *) (source :: * -> * -> *)
       (target :: k -> k -> k) (t :: k -> *) (a :: k) (b :: k).
Semimonoidal p source target t =>
p (source (t a) (t b)) (t (target a b))
forall (t :: * -> *) a b.
Semimonoidal (<--) (:*:) (:*:) t =>
(t a :*: t b) <-- t (a :*: b)
multiply @(<--) @(:*:) @(:*:) in
		(((t :. (u :. t')) := a) -> (:=) (t <:<.>:> t') u a
forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *)
       (t' :: k -> k) (u :: k -> k) (a :: k).
((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a
TUT (((t :. (u :. t')) := a) -> (:=) (t <:<.>:> t') u a)
-> (t ((:.) u t' b) -> (:=) (t <:<.>:> t') u b)
-> (((t :. (u :. t')) := a) :*: t ((:.) u t' b))
-> (:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b
forall (left :: * -> * -> *) (right :: * -> * -> *)
       (target :: * -> * -> *) (v :: * -> * -> *) a b c d.
Bivariant left right target v =>
left a b -> right c d -> target (v a c) (v b d)
<-> t ((:.) u t' b) -> (:=) (t <:<.>:> t') u b
forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *)
       (t' :: k -> k) (u :: k -> k) (a :: k).
((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a
TUT) ((((t :. (u :. t')) := a) :*: t ((:.) u t' b))
 -> (:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b)
-> (((t :. (u :. t')) := a) :*: t ((:.) u t' b))
-> (:=) (t <:<.>:> t') u a :*: (:=) (t <:<.>:> t') u b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ t ((:.) u t' a :*: (:.) u t' b)
-> ((t :. (u :. t')) := a) :*: t ((:.) u t' b)
forall a b. t (a :*: b) -> t a :*: t b
f (u (t' a :*: t' b) -> (:.) u t' a :*: (:.) u t' b
forall a b. u (a :*: b) -> u a :*: u b
g (u (t' a :*: t' b) -> (:.) u t' a :*: (:.) u t' b)
-> t (u (t' a :*: t' b)) -> t ((:.) u t' a :*: (:.) u t' b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>- (t' (a :*: b) -> t' a :*: t' b
forall a b. t' (a :*: b) -> t' a :*: t' b
h (t' (a :*: b) -> t' a :*: t' b)
-> ((t :. (u :. t')) := (a :*: b)) -> t (u (t' a :*: t' b))
forall (t :: * -> *) (u :: * -> *) (category :: * -> * -> *) a b.
(Covariant category category u, Covariant category category t) =>
category a b -> category (t (u a)) (t (u b))
-<$$>- (t :. (u :. t')) := (a :*: b)
xys)) where

instance (Covariant (->) (->) t, Covariant (->) (->) u, Semimonoidal (<--) (:*:) (:*:) t, Semimonoidal (<--) (:*:) (:*:) t', Monoidal (<--) (->) (:*:) (:*:) u, Adjoint (->) (->) t t') => Monoidal (<--) (->) (:*:) (:*:) (t <:<.>:> t' := u) where
	unit :: Proxy (:*:) -> (Unit (:*:) -> a) <-- (:=) (t <:<.>:> t') u a
unit Proxy (:*:)
_ = ((:=) (t <:<.>:> t') u a -> One -> a)
-> Flip (->) (One -> a) ((:=) (t <:<.>:> t') u a)
forall (v :: * -> * -> *) a e. v e a -> Flip v a e
Flip (((:=) (t <:<.>:> t') u a -> One -> a)
 -> Flip (->) (One -> a) ((:=) (t <:<.>:> t') u a))
-> ((:=) (t <:<.>:> t') u a -> One -> a)
-> Flip (->) (One -> a) ((:=) (t <:<.>:> t') u a)
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ \(TUT (t :. (u :. t')) := a
xys) -> (\One
_ -> (u (t' a) -> t' a
forall (t :: * -> *) a. Extractable_ t => t a -> a
extract (u (t' a) -> t' a) -> ((t :. (u :. t')) := a) -> a
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (u :: * -> *) a b.
Adjoint source target t u =>
target a (u b) -> source (t a) b
|-) (t :. (u :. t')) := a
xys)

instance (Covariant (->) (->) t, Covariant (->) (->) t', Adjoint (->) (->) t' t, Bindable (->) u) => Bindable (->) (t <:<.>:> t' := u) where
	a -> (:=) (t <:<.>:> t') u b
f =<< :: (a -> (:=) (t <:<.>:> t') u b)
-> (:=) (t <:<.>:> t') u a -> (:=) (t <:<.>:> t') u b
=<< (:=) (t <:<.>:> t') u a
x = ((t :. (u :. t')) := b) -> (:=) (t <:<.>:> t') u b
forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *)
       (t' :: k -> k) (u :: k -> k) (a :: k).
((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a
TUT (((t :. (u :. t')) := b) -> (:=) (t <:<.>:> t') u b)
-> ((t :. (u :. t')) := b) -> (:=) (t <:<.>:> t') u b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ (((:=) (t <:<.>:> t') u b -> (t :. (u :. t')) := b
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run ((:=) (t <:<.>:> t') u b -> (t :. (u :. t')) := b)
-> (a -> (:=) (t <:<.>:> t') u b) -> a -> (t :. (u :. t')) := b
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. a -> (:=) (t <:<.>:> t') u b
f (a -> (t :. (u :. t')) := b) -> t' a -> u (t' b)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (u :: * -> *) a b.
Adjoint source target t u =>
target a (u b) -> source (t a) b
|-) (t' a -> u (t' b)) -> u (t' a) -> u (t' b)
forall (source :: * -> * -> *) (t :: * -> *) a b.
Bindable source t =>
source a (t b) -> source (t a) (t b)
=<<) (u (t' a) -> u (t' b)) -> t (u (t' a)) -> (t :. (u :. t')) := b
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>- (:=) (t <:<.>:> t') u a -> Primary ((t <:<.>:> t') := u) a
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (:=) (t <:<.>:> t') u a
x

instance (Covariant (->) (->) t, Covariant (->) (->) u, Covariant (->) (->) t', Semimonoidal (->) (:*:) (:*:) t, Semimonoidal (->) (:*:) (:*:) t', Monoidal (->) (->) (:*:) (:*:) u, Adjoint (->) (->) t' t) => Monoidal (->) (->) (:*:) (:*:) (t <:<.>:> t' := u) where
	unit :: Proxy (:*:) -> (Unit (:*:) -> a) -> (:=) (t <:<.>:> t') u a
unit Proxy (:*:)
_ Unit (:*:) -> a
f = ((t :. (u :. t')) := a) -> (:=) (t <:<.>:> t') u a
forall (t :: * -> *) a. Interpreted t => Primary t a -> t a
unite (((t :. (u :. t')) := a) -> (:=) (t <:<.>:> t') u a)
-> (One -> (t :. (u :. t')) := a) -> One -> (:=) (t <:<.>:> t') u a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. (t' a -> u (t' a)
forall (t :: * -> *) a.
Monoidal (->) (->) (:*:) (:*:) t =>
a -> t a
point (t' a -> u (t' a)) -> a -> (t :. (u :. t')) := a
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (u :: * -> *) a b.
Adjoint source target t u =>
source (t a) b -> target a (u b)
-|) (a -> (t :. (u :. t')) := a)
-> (One -> a) -> One -> (t :. (u :. t')) := a
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. One -> a
Unit (:*:) -> a
f (One -> (:=) (t <:<.>:> t') u a) -> One -> (:=) (t <:<.>:> t') u a
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ One
One

instance (Adjoint (->) (->) t' t, Extendable (->) u) => Extendable (->) (t' <:<.>:> t := u) where
	(:=) (t' <:<.>:> t) u a -> b
f <<= :: ((:=) (t' <:<.>:> t) u a -> b)
-> (:=) (t' <:<.>:> t) u a -> (:=) (t' <:<.>:> t) u b
<<= (:=) (t' <:<.>:> t) u a
x = ((t' :. (u :. t)) := b) -> (:=) (t' <:<.>:> t) u b
forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *)
       (t' :: k -> k) (u :: k -> k) (a :: k).
((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a
TUT (((t' :. (u :. t)) := b) -> (:=) (t' <:<.>:> t) u b)
-> ((t' :. (u :. t)) := b) -> (:=) (t' <:<.>:> t) u b
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ (((:=) (t' <:<.>:> t) u a -> b
f ((:=) (t' <:<.>:> t) u a -> b)
-> (t' (u (t a)) -> (:=) (t' <:<.>:> t) u a) -> t' (u (t a)) -> b
forall (m :: * -> * -> *) b c a.
Semigroupoid m =>
m b c -> m a b -> m a c
. t' (u (t a)) -> (:=) (t' <:<.>:> t) u a
forall (t :: * -> *) a. Interpreted t => Primary t a -> t a
unite (t' (u (t a)) -> b) -> u (t a) -> t b
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (u :: * -> *) a b.
Adjoint source target t u =>
source (t a) b -> target a (u b)
-|) (u (t a) -> t b) -> u (t a) -> u (t b)
forall (source :: * -> * -> *) (t :: * -> *) a b.
Extendable source t =>
source (t a) b -> source (t a) (t b)
<<=) (u (t a) -> u (t b)) -> t' (u (t a)) -> (t' :. (u :. t)) := b
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) a b.
Covariant source target t =>
source a b -> target (t a) (t b)
-<$>- (:=) (t' <:<.>:> t) u a -> Primary ((t' <:<.>:> t) := u) a
forall (t :: * -> *) a. Interpreted t => t a -> Primary t a
run (:=) (t' <:<.>:> t) u a
x

instance (Adjoint (->) (->) t' t, Distributive (->) (->) t) => Liftable (->) (t <:<.>:> t') where
	lift :: Covariant (->) (->) u => u ~> t <:<.>:> t' := u
	lift :: u ~> ((t <:<.>:> t') := u)
lift u a
x = ((t :. (u :. t')) := a)
-> TUT Covariant Covariant Covariant t t' u a
forall k k k k k k (ct :: k) (ct' :: k) (cu :: k) (t :: k -> *)
       (t' :: k -> k) (u :: k -> k) (a :: k).
((t :. (u :. t')) := a) -> TUT ct ct' cu t t' u a
TUT (((t :. (u :. t')) := a)
 -> TUT Covariant Covariant Covariant t t' u a)
-> ((t :. (u :. t')) := a)
-> TUT Covariant Covariant Covariant t t' u a
forall (m :: * -> * -> *) a b. Category m => m (m a b) (m a b)
$ (forall a. Category (->) => a -> a
forall (m :: * -> * -> *) a. Category m => m a a
identity @(->) (t' a -> t' a) -> a -> t (t' a)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (u :: * -> *) a b.
Adjoint source target t u =>
source (t a) b -> target a (u b)
-|) (a -> t (t' a)) -> u a -> (t :. (u :. t')) := a
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (u :: * -> *) a b.
(Distributive source target t, Covariant source target u) =>
source a (t b) -> target (u a) (t (u b))
-<< u a
x

instance (Adjoint (->) (->) t t', Distributive (->) (->) t') => Lowerable (->) (t <:<.>:> t') where
	lower :: Covariant (->) (->) u => (t <:<.>:> t' := u) ~> u
	lower :: ((t <:<.>:> t') := u) ~> u
lower (TUT (t :. (u :. t')) := a
x) = (forall a. Category (->) => a -> a
forall (m :: * -> * -> *) a. Category m => m a a
identity @(->) (t' a -> t' a) -> u (t' a) -> t' (u a)
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (u :: * -> *) a b.
(Distributive source target t, Covariant source target u) =>
source a (t b) -> target (u a) (t (u b))
-<<) (u (t' a) -> t' (u a)) -> ((t :. (u :. t')) := a) -> u a
forall (source :: * -> * -> *) (target :: * -> * -> *)
       (t :: * -> *) (u :: * -> *) a b.
Adjoint source target t u =>
target a (u b) -> source (t a) b
|- (t :. (u :. t')) := a
x