Products and Hom-sets

Product

Given a category and collection of objects the product is an object with morphisms such that for all and collection of morphisms , there exists a unique morphism such that the following diagram commutes

Coproduct

Given a category and collection of objects the coproduct is an object with morphisms such that for all and colection of morphisms , there exists a unique morphism such that and the following diagram commutes

Hom-functor

Let be a category and . Define the covariant hom-functor which sends to and to , .

Letting , the contravariant hom-functor sends to and each arrow to the function ,

Show duality of product and coproduct in homset

Let be a category. Denote as the coproduct of and the product of . Show that the following hom-functors are isomorphic

We will prove the equivalence by showing there is a natural isomorphism. For convenience, denote and .

(1) Let be an arrow in . For the coproduct , we have inclusion maps . Given a product , we have morphisms . For all , we can generate a collection of arrows . By the universal property of product, there exists unique such that . We construct similarly. Taking the product , we have the following commutative diagram.

By universal property of coproduct, for all there is a unique such that . Thus our natural transformation is isomorphic and .

(2) Given an arrow , there is a corresponding collection of arrows . There is a unique such that , where is the morphism for . The collection of arrows is in bijection with , thus is a natural isomorphism and .

Show ring homomorphism induces homomorphism between

There are functors and from to . Show that there is a natural transformation from to .

For a commutative ring , and

Consider the natural transformation . Let be an arrow of , so we have the following diagram

is an arrow taking a matrix of to a matrix of component wise. takes the units of to the units of . gives the determinant of matrices in , and likewise for . This diagram commutes, thus is a natural transformation.