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where Φ M,V (ϕ) is the K-linear map defined by prescribing values in V on the basis of
F (M) using ϕ and extending linearly:
Φ M,V (ϕ)( ∑ m∈M
λ m m) := ∑ m∈M
λ m ϕ(m) .
In particular, we find the isomorphism of sets Hom Set (∅, U(V )) ∼ = Hom K (F (∅), V ) for all
K-vector spaces V . Thus Hom K (F (∅), V ) has exactly one element for any vector space
V . This shows F (∅) = {0}, i.e. the vector space freely generated by the empty set is the
zero-dimensional vector space.
2. In general, freely generated objects are obtained as images under left adjoints of forgetful
functors. It is, however, not true that any forgetful functor has a left adjoint. As a counterexample,
take the forgetful functor U from fields to sets. Suppose a left adjoint exists
and study the image K of the empty set under it. Then K is a field such that for any
other field L, we have a bijection
Hom F ield (K, L) ∼ = Hom Set (∅, U(L)) ∼ = ⋆ .
Since morphisms of fields are injective, such a field K would be a subfield of any field L.
Such a field does not exist.
To make contact with the notion of duality, the following reformulation is needed:
Observation 2.5.21.
1. Let F ⊣ G be adjoint functors. From the definition, we get isomorphisms
and
Hom C (G(d), G(d)) ∼ = Hom D (F (G(d)), d)
Hom D (F (c), F (c)) ∼ = Hom C (c, G(F (c))) .
The images of the identity on G(d) and F (c) respectively form together natural transformations
ɛ : F ◦ G → id D and η : id C → G ◦ F .
Note the different order of the functors F, G in the composition and compare to the
definition of a duality.
These natural transformations have the property that for all objects c in C and d in D
the morphisms
G(d) η G(d)
−→ (GF ) G(d) = G(F G)(d) G(ɛ d)
−→ G(d)
and
F (c) F (ηc)
−→ F (GF )(c) = (F G)F (c) ɛ F (c)
−→ F (c)
are identities. Again compare with the properties of dualities. In particular, the left adjoint
of an endofunctor is its left dual in the monoidal category of endofunctors. For proofs, we
refer to [McL, Chapter IV]
2. Conversely, we can recover the adjunction isomorphisms Φ c,d from the natural transformations
ɛ and η by
and their inverses by
Hom C (c, G(d)) → F Hom D (F (c), F (G(d)) (ɛ d) ∗
→ HomD (F (c), d)
Hom D (F (c), d) → G Hom C (G(F (c)), G(d)) η∗ d
→ Hom C (c, G(d)) .
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