Let $X_i,Y_i$ be $S-$schemes with $f:X_1\to X_2$ and $g:Y_1\to Y_2$.
Clearly, this gives rise to $f\times_Sg:X_1\times_S Y_1\to X_2\times_S Y_2$ as $S-$morphism.
Prop 1.33. (2) $f\times_Sg=id_{X_2}\times_Sg\circ f\times_Sid_{Y_1}$ (3) $X\times_SY\to S\times_S Y=Y$ is projection.
$\textbf{Q:}$ Then the book says "Thus, any $f\times_S g$ can be written as composition of projection morphisms." Why $f\times_Sg$ are projection maps? Consider $X_2\times_SY_1\to X_2\times_S Y_2$ given by $Id_{X_2}\times_S g$. This is not projection in general if $X_2$ is not $S$.
Ref. Iitaka, Algebraic Geometry, Chpt 1, Sec 20, Prop 1.33