 We note $\bar{\delta}(w) : \mathcal{Q}\rightarrow \mathcal{Q}$ the function that associates $q$ with $\delta(q,w)$, which we call the effect of $w$. For the sake of simplicity we identify a letter $c$ with its effect $\bar{\delta}(c)$. To maintain whether the contents of the window belong to the language, it suffices to maintain the effect of the window, because a word $w$ belongs to $\mathcal{L}$ iff $(\bar{\delta}(w))(q_0) \in \mathcal{F}$. 