Proposition 4

Suppose that X and Y are compact. Let us identify the space X (resp. the space Y) with $\mathsf{X}(\mathscr{C}(X))$ (resp. $\mathsf{X}(\mathscr{C}(Y))$) (corollary 3 of I, p. 33).

a) For every continuous mapping $\varphi : X\rightarrow$ Y, the mapping $\varphi^*:f\mapsto f\circ \varphi$ is an algebra morphism of $\mathscr{C}(Y)$ into $\mathscr{C}(X)$;

b) The mappings $\varphi \mapsto \varphi^*$ and $\pi \mapsto \mathsf{X}(\pi )$ are inverse bijections between the set of continuous mappings of X into Y and the set of algebra morphisms of $\mathscr{C}(Y)$ into $\mathscr{C}(X)$.