Question on Stein's book thm. 5.1 (homotopical version of Cauchy's theorem)

Question

I have seen other questions about the same proof dealing with different issues but what I can't explain is the following: Let $F(s,t):[0,1]\times[a,b]\to\Bbb{C}$ a continuous function (specifically is a homotopy but it doesn't matter). Since $[0,1]\times[a,b]$ is compact, $F$ is uniformly continuous. Let $\epsilon>0$. In the book it says that there is a $\delta>0$ such that, $$if\, |s_1-s_2|<\delta\,\Rightarrow \sup_{t\in[a,b]}|F(s_1,t)-F(s_2,t)|<\epsilon$$ My problem is the following; to me uniform continuity applied on $F$ would mean that we would have a $\delta$ such that, if $\sqrt{(s_1-s_2)^2+(t_1-t_2)^2}<\delta$ then $|F(s_1,t_1)-F(s_2,t_2)|<\epsilon$. In particular, I feel that uniform continuity would impose restrictions on both of the parameters, not just $s$. Any explanation would be appreciated.