The category of finite sets and Cartesian closed categories
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Applying methods of the proof theory, it is shown that two canonical morphisms are equal in all Cartesian closed categories if and only if some of their realizations in the category of finite sets are equal. All realizations of formal combinations of objects using the functors x and hom are isomorphic in all Cartesian closed categories if and only if some of their realizations in the category of finite sets are isomorphic. On the base of these results, a purely syntactic decision algorithm for (extensional) isomorphism of formal combinations of objects and a new decision algorithm for equality of canonical morphisms are obtained.
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