One can create a Functor between the Vect Category and their Vector Space Duals (i.e. their respective Hom-Sets to their underlying Field $K$ ).

That is, given Vector Spaces $V \to f W$ related by a morphism $f$ , we seek to know the form of $F (f)$ . As it turns out,

F (f) : W^{*} \to V^{*}

not the other way around, which makes $F$ a Contravariant Functor. We denote $F (f) \equiv f^{*}$ : it is called $f$ ‘s Pullback.

The way $F (f)$ acts is as follows: For any Linear Operator $φ : W \to K$ , we have that

f^{*} (φ) = φ \circ f : V \to K

Finally, for any $v \in V$ , we have that

f^{*} (φ) (v) = φ (f (v)) \in K

Functorial properties

For any two morphisms

V_{1} \to T_{1} V_{2} \to T_{2} V_{3}

we have that, given some $φ : V_{3} \to K$ ,

F (T_{2} \circ T_{1}) (φ) = (T_{2} \circ T_{1})^{*} (φ) = φ \circ (T_{2} \circ T_{1}) = (φ \circ T_{2}) \circ T_{1} = T_{2}^{*} \circ (φ \circ T_{1}) = (T_{2}^{*} \circ T_{1}^{*}) (φ) = F (T_{2}) \circ F (T_{1}) (φ)

Thus, we have that $F (T_{2} \circ T_{1}) = F (T_{2}) \circ F (T_{1})$ , and $F$ satisfies a functor’s properties.

Nicholas Funari Voltani