 Limits in a comma category of a limit-preserving functor

If $G:\mathcal{C}\to \mathcal{D}$ preserves small categorical limits, the projection $(X\downarrow G)\to \mathcal{C}$ creates the small categorical limits available in $\mathcal{C}$. A categorical cone of arrows $X\to G{A}_j$ induces a unique arrow $X\to G(\lim A_j)$ by preservation; this equips the underlying categorical limit with its comma category structure. The universal property proves that every categorical cone factorization preserves that structure.