In the context of $V$-enriched category theory, for $F,G : C \to D$ two $V$-enriched functors between $V$-enriched categories, the collection of natural transformations from $F$ to $G$ can also be given the structure of an object in $V$, so that the functor category, denoted $[C,D]$ in the enriched context, is itself a $V$-enriched category.
For $C$ and $D$ $V$-enriched categories, there is a $V$-enriched category denoted $[C,D]$ whose
objects are the $V$-functors $F : C \to D$
hom-objects$\text{ }[C,D](F,G)$ between $V$-functors $F, G$ are given by the $V$-enriched end
over the functor
Write in the following $E_c : [C,D](F,G) \to D(F(c),G(c))$ for the canonical morphism out of the end (the counit).
the composition operation
is the universal morphism into the end $[C,D](K,F)$ obtained from observing that the composites
form an extraordinary $V$-natural family, equivalently that
equalizes the two maps appearing in the equalizer definition of the end.
For $V =$Set, so that $V$-enriched categories are just ordinary locally small categories, the $V$-enriched functor category coincides with the ordinary functor category. (See the examples below.)
To understand the role of the end here, it is useful to spell this out for the case where $V =$ Set, where we are dealing with ordinary locally small categories.
So let $V = Set$ where set is equipped with its cartesian monoidal structure.
Recall the definition of the end over
as an equalizer: it is the universal subobject
of the product of all hom-sets in $D$ between the images of objects in $C$ under the functors $F$ and $G$. So one element $\eta \in \int_{c \in C} D(F(c), G(c))$ is a collection of morphisms
such that the “left and right action” (in the sense of distributors) of $D(F(-),G(-))$ on these elements coincides. Unwrapping what this action is (see the details at end) one find that
the “right action” by a morphism $c \stackrel{f}{\to} d$ is the postcomposition $(F(c) \stackrel{\eta_c}{\to} G(c)) \mapsto (F(c) \stackrel{\eta_c}{\to} G(c) \stackrel{G(f)}{\to} G(d))$
the “left action” by a morphism $c \stackrel{f}{\to} d$ is the precomposition $(F(d) \stackrel{\eta_d}{\to} G(d)) \mapsto (F(c) \stackrel{F(f)}{\to} F(d) \stackrel{\eta_d}{\to} G(d) )$.
So the invariants under the combined action are those $\eta$ for which for all $f : c \to d$ in $C$ the diagram
commutes. Evidently, this means that the elements $\eta$ of the end $\int_{c \in C} D(F(c), G(c))$ are precisely the natural transformations between $F$ and $G$.
An original article is
of the Midwest Category Seminar III (Lecture Notes in Mathematics, Volume 106, 1969), 178 - 191.
Review includes
Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics Vol. 137. Springer-Verlag, 1970, pp 1-38 (pdf)
Max Kelly, section 2.2 p. 29 of Basic concepts of enriched category theory (pdf)
Last revised on January 17, 2019 at 05:38:06. See the history of this page for a list of all contributions to it.