Let be the category whose objects are all finite dimensional vector spaces over a field F

Let ν be the category whose objects are all finite dimensional vector spaces over a field F (of characteristic ≠ 2,3) and whose morphisms are all vector-space isomorphisms. Consider the dual space V* of a left vector space V as a left vector space.

(a) If ϕ: V → V₁ is a vector-space isomorphism (morphism of 'V), then so is the dual map ϕ̅,: V₁* - V* . Hence ϕ̅^-1 : V* → V₁* is also a morphism of ν.

(b) D: ν → ν is a covariant functor, where D(V) = V* and· D(ϕ) = ϕ̅^-1•

(c) For each ν in ν choose a basis { x₁, ... , X_n} and let {∫_x1, •.• , ∫_xn} be the dual bases of V*. Then the map α_v : V → V* defined by x_i| → ∫_x. is an isomorphism. Thus α_v : V ≅ D(V). 

(d) The isomorphism α_v is not natural; that is, the assignment V|→ α_v is not a natural isomorphism from the identity functor I_ν to D.