I believe I've found a typo in this text, but it's also possible that I'm not understanding the statement:
The statement is about transversality. "Suppose $N,M,$ and $S$ are smooth manifolds and for each $s\in S$ we are given a map $F_s:N\rightarrow M$. The collection $\{F_s|s\in S\}$ is called a smooth family of maps if the map $F:M\times S\rightarrow N$ defined by $F(x,s)=F_s(x)$ is smooth."
Shouldn't $F:N\times S\rightarrow M$? Since $F_s:N\rightarrow M$ then $F(x,s)=F_s(x)$ means that $x\in N$, not $x\in M$... am i missing something or is this indeed a typo? I am pretty confident that it's a typo, but since I'm no expert on the material, I want to make sure I'm not missing something...
