Section NTComposeT.
Variable C : PreCategory.
Variable D : PreCategory.
Variables F F' F'' : Functor C D.
Variable T' : NaturalTransformation F' F''.
Variable T : NaturalTransformation F F'.
Let CO := fun c => T' c o T c.
Definition NTComposeT_Commutes s d (m : Morphism C s d)
: CO d o MorphismOf F m = MorphismOf F'' m o CO s.
admit.
Defined.
Definition NTComposeT
: NaturalTransformation F F''
:= Build_NaturalTransformation F F''
(fun c => T' c o T c)
NTComposeT_Commutes.
End NTComposeT.
Definition NTWhiskerR C D E (F F' : Functor D E) (T : NaturalTransformation F F')
(G : Functor C D)
:= Build_NaturalTransformation (F o G) (F' o G)
(fun c => T (G c))
admit.
Class NTC_Composable A B (a : A) (b : B) (T : Type) (term : T) := {}.
Definition NTC_Composable_term `{@NTC_Composable A B a b T term} := term.
Notation "T 'o' U"
:= (@NTC_Composable_term _ _ T%natural_transformation U%natural_transformation _ _ _)
: natural_transformation_scope.
Local Open Scope natural_transformation_scope.

Lemma NTWhiskerR_CompositionOf C D
(F G H : Functor C D)
(T : NaturalTransformation G H)
(T' : NaturalTransformation F G) B (I : Functor B C)
: NTWhiskerR (NTComposeT T T') I = NTComposeT (NTWhiskerR T I) (NTWhiskerR T' I).