# 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.)