Let $F\colon\mathcal{C}\to\mathcal{D}$ be a functor between small categories.

**Question:** Under what conditions is the induced functor
$$F^*\colon\mathsf{Set}^\mathcal{D}\to\mathsf{Set}^\mathcal{C}$$
a logical functor between presheaf toposes?

(I prefer to avoid contravariance if possible, so the "presheaf toposes" I'm referring to here are $\mathrm{Psh}(\mathcal{C}^{op})=\mathsf{Set}^\mathcal{C}$ and $\mathrm{Psh}(\mathcal{D}^{op})=\mathsf{Set}^\mathcal{D}$.)