The bounds of the integral have no dependence on any of the variables, and hence we may move the differential operator into the integrand,

$$\frac{\delta}{\delta \eta (z)} \int \mathrm{d}^4 y \, S_F (z-y) \eta(y) = \int \mathrm{d}^4 y \, S_F (z-y) \delta^{(4)}(z-y)$$

Evaluating the integral using the standard delta distribution identity, we obtain your result, namely $S_F(z-z)$. In this case, the final answer does not pick a minus sign, even though $\eta$ is Grassmann-valued. See Peskin and Schroeder's text on QFT for a summary of Berezin/Grassmann integration.

This post imported from StackExchange Physics at 2014-04-13 14:30 (UCT), posted by SE-user JamalS