Definition
\begin{split}
\hat\delta ∶ Q × \Sigma^{*} &\longrightarrow Q \
\hat\delta(q, \varepsilon) &:= q \
\hat\delta(q, wa) &:= \delta(\hat\delta(q, w), a)
\end{split}
Definition
\begin{split}
\hat\delta ∶ Q × \Sigma^{*} &\longrightarrow Q \
\hat\delta(q, \varepsilon) &:= q \
\hat\delta(q, wa) &:= \delta(\hat\delta(q, w), a)
\end{split}