Now I have $\displaystyle q \circ \bar{f}$, which is, of course, continuous (given the prior statements) from $\displaystyle X \times [0,1] \rightarrow \Sigma Y$.

So if $\displaystyle p$ denotes the canonical quotient map from $\displaystyle X \times [0,1]$ to $\displaystyle \Sigma X$, I would like to say that $\displaystyle F:\Sigma X \rightarrow \Sigma Y$ where $\displaystyle F=p^{-1} \circ (q\circ \bar{f})$ is the desired induced map. However, it remains to be shown (in order to apply the theorem I'm basically quoting here, Theorem 22.2 in Munkres) that $\displaystyle q\circ \bar{f}$ is constant on each set $\displaystyle p^{-1} ([x,t])$, and this is where I am stuck. Looks like this means that $\displaystyle q\circ \bar{f}$ takes a set of things that are equivalent in $\displaystyle X \times [0,1]$ (that's $\displaystyle p^{-1}([x,t])$, of course) and says that these are also equivalent under the quotient map on $\displaystyle Y \times [0,1]$. I can't tie it together.

I've either got the wrong approach, or I could use some intuitive enlightenment. This whole set-up seems to imply some necessary connection between the quotient maps/spaces, but I'm not seeing the "necessitude".