CATEGORIES

X---X Map (Xn, Yn). Again, the mappings (A,- ••,/„) can be represented

as functions on XiU• • • UXn, with (A,••-,/„) |Xt = A-

A pure covariant functor on ^ , • • • , ^

n

is a covariant functor defined on

the product ^ X • • • X-^n- A functor on 5fu • • -, ^n, covariant in the set /

of indices and contravariant in the remaining indices, is a function F on

5fxX ••• X-^to a category Q% taking objects to objects, mappings to map-

pings, identities to identities; taking mappings / = {A}: {X,|—{ Y,} to map-

pings F(f) :F({Zi))-+F(\ Wt)), where for iel, Z,= X, and W^YU butfor

i(£J, Zi= Y,-and Wj=X,; and preserving the composition operation defined

in the product category by gof=h, where /i,=^Afor i £ j , hi=figi otherwise.

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