In the book of Ashok Das, Field theory-path integral approach, he begin the demonstration of the Schwinger-Dyson equation using the fact that the $\delta Z[J]=0$, so \begin{equation} \delta Z[J]=\int \mathcal{D} \phi \frac{\delta S[\phi,J]}{\delta \phi(x)} e^{iS[\phi,J]}=0, \end{equation} but we already know that \begin{equation} \frac{\delta S[\phi, J]}{\delta \phi(x)}=F(\phi(x))-J(x), \end{equation} where $F(\phi(x))$ is the equation of motion.
So if we go back to the first equation and use the identification \begin{equation} \phi(x)\rightarrow -i\frac{\delta}{\delta J(x)}, \end{equation} we conclude that $$ \int \mathcal{D}\phi\left(F(\phi(x))-J(x)\right)e^{iS[\phi,J]}=\left(F\left(-i\frac{\delta }{\delta J(x)}\right)-J(x)\right)\int \mathcal{D}\phi e^{iS[\phi,J]} $$ $$ \left(F\left(-i\frac{\delta }{\delta J(x)}\right)-J(x)\right)Z[J]=\left(F\left(-i\frac{\delta }{\delta J(x)}\right)-J(x)\right)e^{iW[J]}=0. $$
But is here where I get lost, how did he pass from the above equation for $$ e^{-iW[J]}\left(F\left(-i\frac{\delta }{\delta J(x)}\right)-J(x)\right)e^{iW[J]}=F\left(\frac{\delta W[J]}{\delta J(x)}-i\frac{\delta}{\delta J(x)}\right)-J(x)=0. $$
A more important question is: what does this last equation mean at all, mathematically speaking, because the functional derivative now is acting on nothing.