When computing a generating functional, $Z[J]$, in terms of the generating functional of Green functions, $Z[0]$, in my lecturer's notes we reach the following terms:
$$Z[J]= \mathcal{N} \int Dq \hspace{1mm}\text{exp} \left( \frac{-i}{\hbar} \int dt V(q(t))\right)\text{exp}\left(\frac{i}{\hbar} \int ds \hspace{1mm} L_0 (q(t), \dot{q}(t)) + J(s) q(s)\right)\tag{1}$$
$$Z[J]= \mathcal{N} \int Dq \hspace{1mm}\text{exp} \left( \frac{-i}{\hbar} \int dt V\left(-i\hbar\frac{\delta}{\delta J(t) }\right)\right)\text{exp}(...)\tag{2}$$
Can anyone explain to me the identity used for the substitution, $q(t) = -i\hbar\frac{\delta}{\delta J(t)}$ , as this makes no sense to me.
I believe the $-i\hbar$ is just a convention but I don't understand why does $q(t) \to -i\hbar \frac{\delta}{\delta J(t)}$?