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Tom---I understand your general point to be that 2-categories are different,
and from this I tentatively suspect that you would not favor my habit of
calling "horizontal" composition function composition if that proposition were
before the board. I have to give the particulars of this some thought, but while I
am thinking, I'm going to wish that you had been aware of my perspective on
2-categories when you made your comment. What follows is a failed attempt to
convey this perspective in a reasonably brief email in hopes that you will
kindly make some additional comments with this background in view. Reading this
will require some patience, because my lack of erudition is going to show up
here, but let me state that the objective is to define a general natural
transformation to be a functor into a cell category which is actually the first
participating category in a certain type of split interchange category, then to
define its arithmetic in these terms. This is an extremely general but surely not
unprecedented definition of naturality which provides a correspondingly general
definition of "vertical" or (as I prefer) pointwise composition without any
conflicts which I can see, although it is true that function composition does
not seem to exist in this generality. Your sharpest criticisms are very
welcome.

To begin, everyone knows that a double category is an ordered pair of
participating categories which have the same underlying set (of morphisms), and a
double functor is a function between underlying sets which is functorial between
first participants and also between second participants. Say that a double
category splits if the domain and codomain function of each of these participants
is endofunctorial on the other participant. First fact: In this case, the set
of objects of each participant forms a subcategory of the other
participant---call it the object subcategory of the other participant and be careful to
distinguish it from the subcategory of objects which any category has. My excuse
for using "split" in this context is that a category participates with itself
in a split double category exactly when it is a disjoint union of monoids.
Everyone also knows the interchange law for double categories: The compositions
of the participants commute with each other in the weak sense that $ab\#
cd=(a\#c)(b\#d)$ if both sides are defined (visible composition symbols are used as
delimiters in the obvious fashion). Say that a double category is a split
interchange category if it splits and satisfies this interchange law. To be brief,
call it a splintor.

A category participates in a splintor with itself exactly when it is
synonymously its own reverse, opposite or dual category, which amounts to being the
disjoint union of a set of commutative monoids. Call it a core splintor, because
every splintor contains a strongly maximal core subsplintor whose underlying
set consists of those elements at which all four object (i.e., domain and
codomain) functions agree, so that the double objects are obviously in the core.
Strongly maximal means that any core splintor which is a subsplintor of the
given splintor is contained in its core. (Incidentally, aside from core
splintors, I know of only one general type of splintor which has a nondiscrete
core---namely splintors of classical natural transformations under pointwise and
function composition. In fact, in this example, the core consists of those natural
transformations which intertwine identity functors. If an identity functor is
the identity functor of a monoid, the natural transformations which intertwine
it with itself is isomorphic to the classical monoidal center by evaluation
at the monoid's object, and from this I have picked up the habit of saying that
a core component monoid is the center of its object.)

There are a couple of other ways to come across splintors. The easiest is to
just strip off the composition of a category---this gives the discrete, say,
first participant of a splintor for which the second participant is just the
given category itself, so that every category participates in a splintor of some
kind. Call such a splintor a stripping splintor or strippor for short. As in
the case of core categories, every subsplintor of a strippor is a strippor,
and every splintor contains a strippor which is strongly maximal as a contained
strippor: This strippor is just the discrete subcategory of objects of the
first participant and the object subcategory of the second participant. I call
the originally given splintor an objectification of this latter object
subcategory, since it amounts to a way of converting the morphisms of the object
subcategory into the objects of the first participant.

Objectifications are good, because they give a systematic way of converting
the objectified category into a category of functors under function
composition, thus generalizing the Cayley Representation Theorem for groups in a fairly
grandiose manner. This would not lead anyone to think that there would be any
point to objectifying a category which is already discrete, but such
objectifications are precisely the splintors whose second participant's objects are the
splintor double objects. Because of the endofunctoriality of the second
participant's object functions, the homsets of the second participant are
subcategories of the first participant, and bicomposition---simultaneously composing on
the left by one morphism and on the right by another---defines a homset
structuring bifunctor. For this reason, I call objectifications of discrete
categories structuring categories or just structors. This will disgust you, because
structors are what everyone else calls 2-categories. At any rate, every
subsplintor of a structor is a structor, and every splintor contains a structor which
is strongly maximal in the splintor vis-a-vis being a structor. Core
categories and strippors are structors. For that matter, so is a strict monoidal
category, which is just a splintor whose second participant (say) is a monoid.

Here is the crucial property as far as "vertical" or pointwise composition of
natural transformations is concerned. One knows that the functions from a set
into the underlying set of a category have a categorical pointwise
composition: (fg)(xy)=f(x)g(y) when the right side is always defined. So fix a category
and a splintor and consider the functors from the category into the splintor's
first participant. The underlying functions of these functors are stable
under pointwise composition in the second participant, and thus the functors
themselves may be said to form a category under pointwise composition in that
second participant. This is why the homomorphisms from a group into a commutative
group form a group under pointwise composition---because the commutative group
participates in a core splintor with itself. I would almost be willing to say
that a hypergeneral natural transformation is a functor into a splintor first
participant just because you get one of the primary operations of the
arithmetic in this way, but I realistically know that this much generality isn't going
to go far in terms of my talents; so there is a need for more specialized
splintors which more visibly include the classical natural transformation
concept.

This strong market for splintors of various sorts necessitates a more
categorical phrasing of the standard banalities on transitive relations. Given a set
X, define transition composition on its self-cartesian product by
(a,b)(b,c)=(a,c). Any subcategory of this is a transition category on X; the whole thing
is the full transition category X* on X. A transition category is a transitive
relation if it is reflexive in this full transition category; i.e., it has the
same objects. The term "preorder" is dropped. A transitive relation is an
equivalence relation if it is its own subcategory of isomorphisms; it is a
partial ordering if this subcategory is discrete. A function h:X->Y defines a
functor h* between full transition categories by slotwise evaluation: h*(a,
b)=(h(a),h(b)). Every functor between transitive relations is obtained by restricting
and narrowing some such h*. Every equivalence relation is the kernel of some
h*, meaning that it is the inverse image of the discrete subcategory of objects
of the codomain category of h*.

This said, the full transition category of the underlying set of a category
participates with the self-product of the category in a splintor. A subsplintor
for which the first participant is a transitive relation on this underlying
set is a stable transitive relation on the given category. So a partially
ordered group is a splintor. Also interesting is the product category whose first
component is the said full transition category and whose second component is
the said self-product, since it contains various subcategories of "commutative
squares", where I use quotes because I may be referring to commuting to within
an isomorphism or to within an inequality or, generally, to within a morphism
of some specified category which I'll call the value category. I'm now pretty
close to the ideas of a cell category and a cell splintor.

These are splintor concepts. Begin with an objectification (B,C) of the
category A (with composition \# on C and hence on A) whose quasi-commutative
squares are to be constructed. Form the product category A*\times(A\times A), where
A* is the full transition category of the underlying set of A. Take the value
category B to be the first participant of the given objectification, and form
the set [A*\times(A\times A)]\times B, showing no interest in its cartesian
product composition, because there is a subset S of it which has a more
interesting cell composition. To bring this out, write the quintuples in
[A*\times(A\times A)]\times B in attachment form, so that a member looks like (q,u,b,v,p)
with b in B, (u,v) in A\times A and the transition (q,p) in A*. In this, (
q,u,v,p) is the square (of A-morphisms) which is to be regarded as commuting to
within the morphism b. So S consists of those quintuples for which q\# u and v\#
p are defined in A, while b is in the homset of B-morphisms from v\# p to q\#
u, and domains and codomains are organized as follows: The domain of b in C is
the domain of p in C which is also the domain of u in C, while the codomain
of b in C is the codomain in C of q and also the codomain in C of v. These
quintuples are the cells of (B,C). The cell composite of a cell (r,s,c,t,q) with
cell (q,u,b,v,p) is the cell (r,s\#u,(c\#u)(t\#b),t\#v,p). The outside
components are just the composite of (r,s,t,q) with (q,u,v,p) in [A*\times(A\times A)]
when the members of this category are written as attached pairs. The middle
term, which involves one composition in B, is defined whenever the composite
(r,s,t,q)(q,u,v,p) is defined in [A*\times(A\times A)].

So this defines cell composition relative to an objectification. It is
categorical, and the projection (q,u,b,v,p)->(q,u,v,p) is injective when restricted
to the set of objects of S, thus has a subcategory of [A*\times(A\times A)] as
image, and this subcategory is reasonably called the category of squares
which commute to within a B-morphism.

Now I can say that a natural transformation (to within B) is a functor from
some category into the cell category S. To get the idea closer to the classical
form, you notice that following such a functor by the detaching functors
(q,u,b,v,p)->u and (q,u,b,v,p)->v gives candidates for the domain and codomain
functors (my domains are on the right, and codomains are on the left) of the
natural transformation, and, to get a fully extended intertwining function, follow
by (q,u,b,v,p)->b. This last map is functorial into B exactly when the given
splintor (B,C) is a structor (i.e., a 2-category), which is so if and only if
it is surjective. Both B and C have functorial representations in S in this
case, which is my personal, idiosyncratic explanation of why the elements of
2-categories are called cells.

To get pointwise composition out of this, you construct a second
participating cell category by first of all reversing B to get a splintor (B',C) which
still objectifies A. Then you construct the cell category of this semireversed
splintor and apply the double switch (q,u,b,v,p)->(v,p,b,q,u) to pull the
semireverse cell composition back onto the underlying set of the first cell category
S. This gives the second participant T of the the cell splintor (S,T) of
(B,C). Pointwise composition of natural transformations means pointwise
composition of functors into S in T. By the way, (S,T) is a structor exactly when the
objectified category A is discrete, which reflects the fact that forming a cell
splintor does not change (except by a functorial isomorphism) a splintor's
maximal structor, nor does the core change. This is my argument that structors
are not enough to fully describe the cell concept.

To get the classical idea of a natural transformation, begin with a discrete
value category B; that is, begin with the strippor of C, then regard the
natural transformation as running from A to C. This is justified by the fact that,
in this case, the cell category S is isomorphic to its commutation category by
the projection (q,u,b,v,p)->(q,u,v,p); so the intertwining function
(q,u,b,v,p)->b can just as well be written as (q,u,v,p)->qu=vp, and so on. When this
follows the functor version of a classical natural transformation, it gives the
fully extended intertwining function which I mentioned in my first email.

You can see that, as far as this point of view goes, there is no particularly
obvious conflict between function composition of natural transformations and
pointwise composition. Function composition doesn't obviously exist when
values are not discrete. There is presumably still plenty to be said in terms of
function composition of splintor functors, but I haven't thought about this at
all, and I'm not likely to start until I have understood your note.

Thanks for your comments and your patience if you have any left.

Pat Donaly