Example 8.5:

Let $X$ be a topological space and $C(X)$ be the set of all continuous functions from $X$ to the real line (with its usual topology). Then $C(X)$ is a commutative ring with unity. Suppose that $f : X \longrightarrow Y$ is a continuous map between topological spaces then we define $f^{*}$ to be the map

$$f^{*} : C(Y) \longrightarrow C(X)$$

$$\phi\mapsto \phi\circ f\phantom{XX}$$

It is obvious to see that $f^{*}$ is a ring homomorphism and id$_X^{*} =$   id$_{C(X)}$. Further, $(g\circ f)^{*} = f^{*}\circ g^{*}$ for $f \in$   Mor$(X, Y)$ and $g \in$   Mor$(Y, Z)$. We thus have a contravariant functor Top$\longrightarrow$   Rng sending the object $X \in$   Top to the object $C(X) \in$   Rng and assigning to $f \in$   Mor$(X, Y)$ the ring homomorphism $f^{*} \in$   Mor$(C(Y), C(X))$.