Example 1

Let X and Y be topological spaces and $f: X\rightarrow Y$ a continuous mapping. Let $(T, q)$ be a Y-space. Let Z be the X-space $X\times_YT$, endowed with the mapping pr$_1$; let us denote by $\tau$ its canonical descent datum relative to $f$ (IV, p. 383, Example 1). The subsets of Z of the form $X\times_YV$, where V is an open subset of T, are open in Z and saturated for the relation $R_{\tau}($loc. cit.). By definition of the product topology, the sets $U\times_Y$ pr$^{-1}_2(V)$, where U is open in X and V is open in T, form a basis of the topology of $X\times_YZ$. Consequently, the canonical descent datum on a fibred product $X\times_YT$ is effective.

The canonical mapping $Z/R_{\tau}\rightarrow T$ is injective and continuous. It is not, however, necessarily surjective, nor strict (IV, p. 462, exerc. 1).