 Solution

Take any limiting cone ${\ell }_i:L\to D(i)$ in $\mathcal{C}$ of the given shape. Since $GF$ preserves such categorical limits, the cone $GF{\ell }_i:GFL\to GFD(i)$ is limiting in $\mathcal{E}$. It is the image under $G$ of the cone $F{\ell }_i$ in $\mathcal{D}$. Reflection by $G$ therefore says that $F{\ell }_i$ is limiting in $\mathcal{D}$. Since this works for every diagram and every limiting cone of the specified shape, $F$ preserves categorical limits of that shape.