Proposition 3

Let X and Y be topological spaces and let $f: X\rightarrow$ Y be a continuous mapping. Let $(Z, p)$ and $(Z', p')$ be X-spaces endowed with descent data relative to $f$, denoted respectively by $\tau$ and $\tau '$. If the descent datum $\tau '$ is effective, the mapping $\varphi \mapsto \overline{\varphi}$ is a bijection of $\mathscr{C}_{\tau ,\tau'}(Z; Z')$ onto $\mathscr{C}_Y(Z/R_{\tau}; Z'/R_{\tau'})$.

For every X-morphism $\varphi$ of Z into $Z'$ compatible with the descent data, the mapping $\overline{\varphi}$ is a Y-morphism. Conversely, let $g: Z\rightarrow Z/R_{\tau}$ and $g': Z'\rightarrow Z'/R_{\tau'}$ denote the canonical mappings, and let $\psi : Z/R_{\tau}\rightarrow Z'/R_{\tau'}$ be a Y-morphism. The mappings $p: Z\rightarrow X$ and $\psi \circ g: Z\rightarrow Z'/R_{\tau'}$ are Y-morphisms. The hypothesis that the descent datum $\tau '$ is effective means that the diagram

$p'q'$

X $^f$ Y

is a cartesian square. Hence there exists a unique continuous mapping $\varphi : Z\rightarrow Z'$ such that $p'\circ \varphi =p$ and $g'\circ \varphi =\psi \circ g$. The first equality means that $\varphi$ is an X-morphism, the second equality means that $\varphi$ is compatible with the descent data and that $\overline{\varphi}=\psi$, whence the proposition.