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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