Adjoint Functors - Adjunctions in Full - Universal Morphisms Induce Hom-set Adjunction

Universal Morphisms Induce Hom-set Adjunction

Given a right adjoint functor in the sense of initial morphisms, do the following steps.

  • Construct a functor and a natural transformation .
    • For each object of, choose an initial morphism from to, so we have . We have the map of on objects and the family of morphisms .
    • For each, as is an initial morphism, then factorize with and get . This is the map of on morphisms.
    • The commuting diagram of that factorization implies the commuting diagram of natural transformations, so is a natural transformation.
    • Uniqueness of that factorization and that is a functor implies that the map of on morphisms preserves compositions and identities.
  • Construct a natural isomorphism .
    • For each, as is an initial morphism, then is a bijection, where .
    • is a natural transformation, is a functor, then Phi_{Y_1,X_1}(xcirc fcirc F(y))
= G(x)circ G(f)circ G(F(y))circeta_{Y_1}
= G(x)circ G(f)circ eta_{Y_0}circ y
= G(x)circ Phi_{Y_0,X_0}(f)circ y, then is natural in both arguments.

A similar argument allows one to construct a hom-set adjunction from the terminal morphisms to a left adjoint functor. (The construction that starts with a right adjoint is slightly more common, since the right adjoint in many adjoint pairs is a trivially defined inclusion or forgetful functor.)

Read more about this topic:  Adjoint Functors, Adjunctions in Full

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