Inverse Limit of Finite Direct Product of Groups/TopSpaces

Question

I’m wondering if the inverse limit of a finite direct product of groups $G_{i,j}$ or topological spaces $X_{i,j}$ is isomorphic to the direct product of the inverse limits of $G_{i,j}$ or $X_{i,j}$, respectively.

I’m thinking that both of these are true but I’m not sure.

Thanks in advance.