• 1. Functoriality. The assignment A,B,(A,B)AB defines functors
    A:SetsSets,B:SetsSets,12:Sets×SetsSets,

    where 12 is the functor where

    • Action on Objects. For each (A,B)Obj(Sets×Sets), we have

      [12](A,B)=defAB.

    • Action on Morphisms. For each (A,B),(X,Y)Obj(Sets), the action on Hom-sets

      (A,B),(X,Y):Sets(A,X)×Sets(B,Y)Sets(AB,XY)

      of at ((A,B),(X,Y)) is defined by sending (f,g) to the function

      fg:ABXY

      defined by

      [fg](x)=def{(0,f(a))if x=(0,a),(1,g(b))if x=(1,b),

      for each xAB.

    and where A and B are the partial functors of 12 at A,BObj(Sets).


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