Let $G$ be a profinite group. As is well-known, we may write $G = \operatorname{lim} G/N$ for $N$ family of normal open compact subgroups of $G$ (which is a cofiltered limit, and we may index it by $\mathbb{N}$). For any pair of such subgroups $N$ and $M$ such that $N \subseteq M$, there is a projection $\pi_{NM} \colon G/N \to G/M$, which induces a functor $\operatorname{Ind}_{G/N}^{G/M} \colon \operatorname{Rep}(G/N) \to \operatorname{Rep}(G/M)$.
This defines a diagram in the category of tensor categories (with maybe well-behaved functors). Is $\operatorname{Rep}(G)$ the limit of this diagram?