Part One
This part improves the results of [11] and [18] by imitating the
method of [11] and [18] as closely as possible. The main result here is
Theorem 5, which says that two allowable natural transformations h and h'
having the same domain, the same codomain and the same graph are equal,
provided the pair (h,h') is not in a certain class Of . We are calling this
an "abstract" coherence theorem, because it is not directly applicable in
practice; given the allowable natural transformations h and h', one cannot
always tell, by just looking at them, whether (h,h') is in Of .
Later, in Part Two, we prove that (h,h')^3T is also a necessary con-
dition for h = h'. It is remarkable that the method of [11] has been pow-
erful enough both to suggest and to prove the full "if" part of an "if and
only if" coherence theorem for closed categories.
In [11],the idea is to prove, by induction on rank and by using the
cut elimination theorem for allowable natural transformations, that cer-
tain pairs (h,h') of allowable natural transformations with Th = Th' have
h = h'. That not all such pairs can have h = h', is established by an ex-
ample ([11,p.100]); Kelly-MacLane prove the theorem for all pairs (h,h')
with |~ h = [~h T , in which the domain and codomain of h and hf are proper
(p.vii). That the class of the (h,h') with improper domains and/or codo-
mains had to be excluded, seems to have been suggested by the details of
the intermediate proofs. In fact, the proof of the final theorem (in [11])
needs Propositions 7.6 and 7.8 ([11,pp.131,133]) and there the hypothesis
that the shapes are proper is used.
Received by the editors June 17,1975 and, in revised form June 4,1976.
Research supported by the National Hellenic Research Foundation.
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