I am starting to work with functional derivatives. In Density functional theory an advanced course by Engel and Dreizler they have the example of the functional in the title of this question. The variation of the functional can be calculated to be
$$\delta F_w = \int_{y_1}^{y_2} dx_1 \int_{y_1}^{y_2} dx_2 w(x_1,x_2)[(f(x_1)\varepsilon \eta(x_2) + f(x_2) \varepsilon \eta(x_1) + \varepsilon \eta(x_1) \varepsilon \eta(x_2)] $$
The Taylor Expansion around $\varepsilon = 0$ is
$$ \delta F_w = \frac{dF(f + \varepsilon \eta)}{d \varepsilon} \bigg|_{\varepsilon = 0} \varepsilon + \frac{d^2F(f + \varepsilon \eta)}{d \varepsilon^2} \bigg|_{\varepsilon = 0} \varepsilon^2= \\= \int_{y_1}^{y_2} dx_1 \int_{y_1}^{y_2} dx_2 w(x_1,x_2)(f(x_1)\eta(x_2) + f(x_2) \eta(x_1)) \varepsilon + \int_{y_1}^{y_2} dx_1 \int_{y_1}^{y_2} dx_2 w(x_1,x_2)2 \eta(x_1) \eta(x_2) \varepsilon^2 $$
By the definition of functional derivative I get that in the first term I need to isolate $\eta(x_1)$, this can be done by rewriting it in the following way
$$\frac{dF(f + \varepsilon \eta)}{d \varepsilon} \bigg|_{\varepsilon = 0} = \int_{y_1}^{y_2} dx_1 \int_{y_1}^{y_2} dx_2 (w(x_1,x_2) + w(x_2,x_1))f(x_2)\eta(x_1)$$
Which means that the first functional derivative is
$$ \frac{\delta F}{\delta f(x_1)} = \int_{y_1}^{y_2} dx_2 (w(x_1,x_2) + w(x_2,x_1))f(x_2)$$
So far, this agrees with the textbook, however there is something I do not understand in the second derivative, and it might be just some simple algebra issue. We have that
$$ \frac{d^2F(f + \varepsilon \eta)}{d \varepsilon^2} \bigg|_{\varepsilon = 0} = \int_{y_1}^{y_2} dx_1 \int_{y_1}^{y_2} dx_2 w(x_1,x_2)2 \eta(x_1) \eta(x_2)$$
From the definition of the second variational derivative I conclude from the above that
$$ \frac{\delta^2 F_w}{\delta f(x_1) \delta f(x_2)} = 2 w(x_1,x_2)$$
However the textbook writes that
$$\frac{\delta^2 F_w}{\delta f(x_1) \delta f(x_2)} = w(x_1,x_2) + w(x_2,x_1)$$
I get that (unless there is something I am not seeing)
$$\int_{y_1}^{y_2} dx_1 \int_{y_1}^{y_2} dx_2 w(x_1,x_2)2 \eta(x_1) \eta(x_2) = \int_{y_1}^{y_2} dx1 \int_{y_1}^{y_2} dx_2 (w(x_1,x_2) + w(x_2,x_1)) \eta(x_1) \eta(x_2) $$
However, $2w(x_1,x_2) \neq w(x_1,x_2) + w(x_2,x_1)$ unless $w(.,.)$ is symmetric, so my solution to the second derivative differs from the one in the textbook. This might be just some simple algebra I am not seeing, what am I missing?