The category of (small) categories and functors is the category Cats where

  • Objects. The objects of Cats are small categories.
  • Morphisms. For each C,DObj(Cats), we have

    HomCats(C,D)=defObj(Fun(C,D)).

  • Identities. For each CObj(Cats), the unit map

    1CCats:ptHomCats(C,C)

    of Cats at C is defined by

    idCCats=defidC,

    where idC:CC is the identity functor of C of Example 9.5.1.1.4.

  • Composition. For each C,D,EObj(Cats), the composition map

    C,D,ECats:HomCats(D,E)×HomCats(C,D)HomCats(C,E)

    of Cats at (C,D,E) is given by

    GC,D,ECatsF=defGF,

    where GF:CE is the composition of F and G of Definition 9.5.1.1.5.


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