Pullback along a family of maps

Question

Is there a construction to handle pullbacks along many maps simultaneously? Something like this: Given a subset $S \subseteq C(B',B)$ and a bundle $\xi = (E,\pi,B)$. Is there a bundle " $S^*\xi$ " ? If $S = \{f\}$ for some $f\in C(B',B)$, it should be the standard pullback $f^*\xi = \{f\}^*\xi$.

Answer 1

You can use the space $S\times B'=\sqcup_{s\in S}B'$ and turn all the morphisms $s\in S$ into one single morphism $S\times B' \overset{\sqcup_s s}\longrightarrow B$, along which you can pull back.