In section 4.1 of Vistoli's notes he starts by showing we may locally construct arrows in $\mathsf{Top}^2$, i.e arrows over $U$.
Then, the author says that we can moreover do the same for spaces, although this is more complicated. This is followed by the proposition below:
Proposition 4.1. Suppose we're given a space $U$ with an open cover $\left\{ U_i \right\}$ and maps $u_i:X_i\rightarrow U_i$ satisfying the cocycle condition. Then there's a continuou map $u:X\rightarrow U$ etc...
Why does the author say this construction is for spaces, and how is it different than locally construct a continuous map?