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Set Implicit Arguments.
Definition admit {T} : T.
Admitted.
Delimit Scope object_scope with object.
Delimit Scope morphism_scope with morphism.
Delimit Scope category_scope with category.
Delimit Scope functor_scope with functor.
Reserved Infix "o" (at level 40, left associativity).
Record PreCategory :=
{ Object :> Type;
Morphism : Object -> Object -> Type;
Compose : forall s d d', Morphism d d' -> Morphism s d -> Morphism s d' where "f 'o' g" := (Compose f g) }.
Bind Scope category_scope with PreCategory.
Infix "o" := (@Compose _ _ _ _) : morphism_scope.
Local Open Scope morphism_scope.
Record Functor (C D : PreCategory) :=
{ ObjectOf :> C -> D;
MorphismOf : forall s d, C.(Morphism) s d -> D.(Morphism) (ObjectOf s) (ObjectOf d);
FCompositionOf : forall s d d' (m1 : C.(Morphism) s d) (m2: C.(Morphism) d d'),
MorphismOf _ _ (m2 o m1) = (MorphismOf _ _ m2) o (MorphismOf _ _ m1) }.
Bind Scope functor_scope with Functor.
Arguments MorphismOf [C%category] [D%category] F%functor [s%object d%object] m%morphism : rename, simpl nomatch.
Definition ComposeFunctors C D E
(G : Functor D E) (F : Functor C D) : Functor C E
:= Build_Functor C E (fun c => G (F c)) admit admit.
Infix "o" := ComposeFunctors : functor_scope.
Record NaturalTransformation C D (F G : Functor C D) :=
{ ComponentsOf :> forall c, D.(Morphism) (F c) (G c);
Commutes : forall s d (m : C.(Morphism) s d),
ComponentsOf d o F.(MorphismOf) m = G.(MorphismOf) m o ComponentsOf s }.
Definition NTComposeT C D (F F' F'' : Functor C D)
(T' : NaturalTransformation F' F'')
(T : NaturalTransformation F F')
(CO := fun c => T' c o T c)
: NaturalTransformation F F''.
exact (Build_NaturalTransformation F F''
(fun c => T' c o T c)
(admit : forall s d (m : Morphism C s d), CO d o MorphismOf F m = MorphismOf F'' m o CO s)).
Defined.
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.

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