COHERENCE AND NON-COMMUTATIVE DIAGRAMS IN CLOSED CATEGORIES 5

fa"1 = gM(f"Bll)x" = g"(f"tSl)u((lSs)S(lSz)) = g*(m81)x*

with g* allowable and x* central. Then

h = 7r(fa_1a) = [1,g*(mEl)x*a]d = [1,g*a]d(m®l)x**

with x** central; thus h is of the form

([Q,M]SP)SN JL+. ([F,G]HEf)BHf

m]S1

. GSH' -i-^ S ,

where E1 = [Q,M]BPE , Hf = PH®N , x" = x** is central, m and g" = [l,g*a]d

are allowable, m cannot be written in the form , since £" (and therefore

ft = b(mSn) ) cannot be so written and, via x", [Q,M] is associated with

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

This contradicts the hypothesis on h, so fa" satisfies the last hypothe-

sis of the present proposition, as needed.

If h is of type , we distinguish cases (i) , (ii) , (iii) as in [11]

and [18] and use the notation of [11] and [18]. By Lemma 1, we may assume

that f cannot be written in the form . By our early assumption, [B,C]

is not constant (notation of [11] and [18]). Case (i) is done as in [18],

observing that, in Subcases II,III,IV, the allowable natural transforma-

tion s, on which the induction hypothesis is used, clearly satisfies (now)

the last hypothesis of the present proposition (instead of etty ), since h

does so. In Case (ii) , Subcase I is excluded because it contradicts the

last hypothesis on h; Subcase II is done as in [18];however, if r(Pf)) = 0,

then the ft (of [11]) satisfies the last hypothesis of the present proposi-

tion since h does so, and therefore, by induction, ft is of the form ,

contrary to our assumption; so the case r(Pn) = 0 is excluded. Case (iii)

is done as in [18]; in Subcase II, that the allowable natural transforma-

tion g(lSt)w (of [11]) satisfies the last hypothesis of the present propo-

sition, as needed now, is proved as follows: Suppose that g(lSt)w can be

written in the form

([Q,M]HP)1S(CBIR) -^- ([F,G]HE)HH f"B1 GSH -&U- S ,

where f" and g" are allowable, x" is central, f" cannot be written in the

form and, via x", [Q,M] is associated with a prime factor of E and

[F,G] is associated with a prime factor of P. Let P^ (resp.PH) be an iter-

ated S-product of the prime factors of P associated via x" with prime fac-

tors of E (resp.H). Similarly define Cp , CH , Rp , RH . There are obvious

centrals