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The functor l i i p r o j F is left adjoint and the functor limind F is right adjoint to the diagonal functor A : C + Funct(J,C). d. Let F be the functor from the category S c a t of small categories (ObC is a set) to the category Set, which associates to each C the set ObC. Then F has the left adjoint functor GI,which associates to a set X the discrete category Cx (ObCx = X , Homc,(x,x) = id,, Homc,(x, y) = 0 if x # y), and the right adjoint functor G,, which associated to X the category with is an oneelement set for all x, y.

For any monomorphism X' + X and any morphism X' + Y there exists a morphism Y --t X that makes the injectivity diagram commutative. We claim that the the statement a (resp. the statement b) is equivalent t o the projectivity (resp. to the injectivity) of the object Y. 8. Projective Modules and Free Modules. The results of the previous subsection yield the following description of projective objects in the category of modules (left-or right): a module is projective if and only if it is a direct summand of a free module.

5 2. Additive and Abelian Categories 35 c. The functor l i i p r o j F is left adjoint and the functor limind F is right adjoint to the diagonal functor A : C + Funct(J,C). d. Let F be the functor from the category S c a t of small categories (ObC is a set) to the category Set, which associates to each C the set ObC. Then F has the left adjoint functor GI,which associates to a set X the discrete category Cx (ObCx = X , Homc,(x,x) = id,, Homc,(x, y) = 0 if x # y), and the right adjoint functor G,, which associated to X the category with is an oneelement set for all x, y.

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A0-stable linear multistep formulas of the-type by Rockswold G. K.


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