How to find out adjoint of the following linear operator?

Question

Consider Hammerstein integral equation $$F: D(F) = H^1[0,1] \to L^2[0,1]$$ $$F(x)(s) = \int_{0}^{s} (s-t) x^3(t)dt$$ Now its Frechet derivative is given by $$(F'(x)h)(s) = 3 \int_{0}^{s} (s-t) x^2(t)h(t)dt$$ Now how to calculate the adjoint of this operator $F'(x)$.