COHERENCE AND NON-COMMUTATIVE DIAGRAMS IN CLOSED CATEGORIES 3

is a modification (and common generalization) of Proposition 708 of [11]

and Proposition C of [18,§1], We first give an easy lemma.

LEMMA 1. If h:T —v S is an allowable natural transformation, which is of

type , being a composite

h:T -iU»([B,C]BA)BD fSalCBD -i^-S ,

with x central and f,g allowable, then h can be written in the form as

h:T - ^ ([X,Y]HZ)HW SB1 YBW —t-^s ,

with u central, s and t allowable and s not expressible in the form .

Proof^ If f is not of type , take u = x, s = f, t = g. If f is of type

, we imitate for allowable natural transformations the argument used in

[11,p.139] for graphs in the analogous situation. •

PROPOSITION 2. Let h:([Q,M]HP)HN - S be an allowable natural transforma-

tion, with [Q,M] non-constant. Suppose that [" h is of the form 7|(SH1) for

graphs £:P —-Q , TpMHN —-S , such that £ cannot be written in the form

p(eBl)w for any graphs p,&,cj with co central. Suppose finally that h can-

not be written in the form

([Q,M]HP)HN X' ([F,G]BE)KH ftS)1 GSH - 1 ^ S ,

where xT,ff,g' are allowable natural transformations, xT is central, fT

cannot be written in the form and, via xf, [Q,M] is associated with a

prime factor of E and [F,G] is associated with a prime factor of P. Then

there are allowable natural transformations p:P —9- Q , q:MSN --S such

that Tp = £ , rq = t\ and h = q(pSl) .

?I99^i We ^° induction on r (h) , using the cut elimination property of al-

lowable natural transformations (pp,xiv,xvi), and following the proof of

Proposition C of [18,§1] and the proof of Proposition 7.8 of [11], re-

placing the hypothesis Th^ of [18] by the last hypothesis of the present

proposition. As in [11],we may assume that P,N and S have no constant

prime factors and we distinguish cases according to the possible forms of h

The case of h being central is done as in [11],

The case of h being of the form y(fSg)x, for allowable f:A — - C and