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A simple example arises in the category of vector spaces. Fix a vector space <math>V</math>. We can define a functor <math>\textrm{Hom}(V,-) : \textrm{Vect}_k \to \textrm{Vect}_k</math> by
A simple example arises in the category of vector spaces. Fix a vector space <math>V</math>. We can define a functor <math>\textrm{Hom}(V,-) : \textrm{Vect}_k \to \textrm{Vect}_k</math> by
#An object <math>A</math> (that is a vector space) is sent to <math>\textrm{Hom}(V,A)</math>, the vector space of linear maps from <math>V</math> to <math>A</math>. We can think of these linear maps as matrices.
#An object <math>A</math> (that is a vector space) is sent to <math>\textrm{Hom}(V,A)</math>, the vector space of linear maps from <math>V</math> to <math>A</math>. We can think of these linear maps as matrices.
#A linear transformation <math> \phi : A \to B</math> (i.e., an element of <math>\textrm{Hom}_{\textrm{Vect}_k}(A,B)</math> is sent to a linear transformation <math>\textrm{Hom}(V,A) \to \textrm{Hom}(V,B)</math> by setting <math>\textrm{Hom}(V,-)(\phi) = (f \mapsto \phi \circ f)</math>. I other words, the image of <math>\phi</math> is supposed to be a linear transformation from <math>\textrm{Hom}(V,A)</math> to <math>\textrm{Hom}(V,B)</math>: it is defined by sending a linear transformation <math>f : V \to A</math> to <math>\phi \circ f : V \to B</math>.
#A linear transformation <math> \phi : A \to B</math> (i.e., an element of <math>\textrm{Hom}_{\textrm{Vect}_k}(A,B)</math> is sent to a linear transformation <math>\textrm{Hom}(V,A) \to \textrm{Hom}(V,B)</math> by setting <math>\textrm{Hom}(V,-)(\phi) = (f \mapsto \phi \circ f)</math>). In other words, the image of <math>\phi</math> is supposed to be a linear transformation from <math>\textrm{Hom}(V,A)</math> to <math>\textrm{Hom}(V,B)</math>: it is defined by sending a linear transformation <math>f : V \to A</math> to <math>\phi \circ f : V \to B</math>.
We can similarly obtain a contravariant function <math>\textrm{Hom}(-,V)</math> by
We can similarly obtain a contravariant function <math>\textrm{Hom}(-,V)</math> by