Working with the word w⋅y, while given the word y⋅w $L$ is a regular language. I am given $F(L)$ such that
$$F(L)= \{wy \mid yw\in L\}$$
I need to prove that if $L$ belongs to $L_\text{dfa}$, $F(L)$ also belongs to $L_\text{dfa}$.
I am having a hard time understanding how to work with words of varying length (both $w$ and $y$ could be of any length, even for the same word $yw$)
How can I use the fact that $L$ belongs to $L_\text{dfa}$? I need to assume things about substrings of words in $L$, and I can't tell if they are valid words or not..

Let $M$ be a dfa accepting $L$. Especially, let $S$ be the set of states of $M$, $s_0\in S$ the initial state and $T\subseteq S$ the accepting states.
For $s_0'\in S$, $T'\subseteq S$ let $M_{s_0',T'}$ denote the dfa obtained from $M$ by taking $s_0'$ as initial and $T'$ as accepting instead and let $L_{s_0',T'}$ be the langugae accepted by $M_{s_0',T'}$. So especially $M=M_{s_0,T}$ and $L=L_{s_0,T}$. Show that
$$ F(L)=\bigcup_{s,t\in S}L_{s,\{t\}}L_{t,\{s\}}$$
and that this allows ouy to constuct a dfa for $F(L)$.
