COHERENCE AND NON-COMMUTATIVE DIAGRAMS IN CLOSED CATEGORIES
(in other words, d and e are natural in A and B in the sense of [5])
(vi) for any objects A,B,C,D of V, the following diagrams are commutative:
CI
((ABB)BC)BD —*- (ABB)B(CBD)
a
AS(BB(CBD))
aBl
(AB(BBC))BD
18a
AB((BBC)BD)
C2
AB(BBI)
18b
C3
ABB
1
ABB
C4
(ABB)BC ^—^AB(BBC) ^--(BBC)BA
cBl
(BBA)BC —•- BB(ABC) =^^ BB(CBA)
C5
C6
^ [B,[B,A]BB]
fl,e]
[B,ABB]BB
ABB
The arrows aABC, a^c , bA , b^1 , cAB , dAB , eAfi of (iv) and (v)
are the components in V of natural transformations (in the sense of [5])
-1 -1
a,a ,b,b ,c,d,e. Any (meaningful) word formed by 8, [,] and natural
transformations, one of which is an element of {a,a ,b,b ,c,d,e} and all
the others are identities, is an expanded instance of the non-identity nat-
ural transformation it involves. The identity natural transformations and
the composites of simple or expanded instances of elements of {a,a ,b,
b ,c,d,e} have been called canonical ([15]) or allowable ([11]) natural
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