Example 8.4:

Here we give an example of a contra-variant functor. The family of all real vector spaces, denoted by ${\bf Vect}$ is a category and for a pair of real vector spaces $V$ and $W$, the set Mor$(V, W)$ consists of all linear transformations from $V$ to $W$. We define a functor from ${\bf Vect}$ to itself by assigning to each $V$ its dual $V^{*}$ and to each $T \in$   Mor$(V, W)$ the adjoint map $T^{*}$. Again,

$$($$id$$_V)^{*} =$$   id$$_{V^{*}}$$

But if $U, V$ and $W$ are three vector spaces and $T \in$   Mor$(U, V)$ and $S\in$   Mor$(V, W)$ are two linear maps then

$$(S\circ T)^{*} = T^{*}\circ S^{*}$$

Let us look at an example of a functor from the category of topological spaces to the category Rng of commutative rings. We shall always assume that every ring that we shall deal with, has a unit element.