structural:expansion, linguistic:translation
Definition via Hom-set adjunction

A hom-set adjunction between two categories C and D consists of two functors F : D → C and G : C → D and a natural isomorphism

This specifies a family of bijections

for all objects X in C and Y in D.

In this situation, F is left adjoint to G and G is right adjoint to F.

This definition is a logical compromise in that it is more difficult to satisfy than the universal morphism definitions, and has fewer immediate implications than the counit–unit definition. It is useful because of its obvious symmetry, and as a stepping-stone between the other definitions.

In order to interpret Φ as a natural isomorphism, one must recognize hom (F –, –) and hom (–, G –) as functors. In fact, they are both bifunctors from D × C to Set (the category of sets). For details, see the article on hom functors. Explicitly, the naturality of Φ means that for all morphisms f : X → X′ in C and all morphisms g : Y ′ → Y in D the following diagram commutes:

The vertical arrows in this diagram are those induced by composition. Formally, Hom(Fg, f) : Hom (FY, X) → Hom (FY′, X′) is given by h → f o h o Fg for each h in Hom (FY, X). Hom(g, Gf) is similar.