For a given language $L$ and $w \in X^{\ast}$ denote by $w^{-1} L := \{ u \in X^{\ast} : wu \in L \}$ the (left) quotients of $L$. The left quotients are closely related to the minimal automaton of $L$ in case $L$ is regular, i.e. in that case $(X, Q, \delta, q_0, F)$ with $Q := \{ w^{-1} L : w \in X^{\ast} \}$, $\delta(w^{-1}L, x) := x^{-1}(w^{-1}L)$,$F := \{ w^{-1}L \in Q : \varepsilon \in w^{-1}L \}$ is isomorphic to the minimal complete automaton accepting $L$.
I am interested in the minimal automaton of a given quotient $w^{-1}L$ in case $L$ is regular, is there a way to construct it? Also as the quotients and the states in a minimal automaton correspond to each other, I am somehow interested in the set $\{ u^{-1}(w^{-1}L) : u \in X^{\ast} \}$ for given quotient $w^{-1}L$.
Note also that the quotients are closely connected to the Nerode (right) congruence, i.e. $u \sim_L v \Leftrightarrow \forall x \in X^{\ast} : ux \in L \leftrightarrow vx \in L \Leftrightarrow u^{-1}L = v^{-1}L$, giving a bijection between them; and also yielding that the minimal automaton of $L$ is an accepting automaton for the nerode classes.