A cofinal subsystem has the same inverse limit up to canonical isomorphism

Restricting a compatible tuple on $I$ to its $J$-coordinates gives a homomorphism Compatibility is preserved because every relation used in the $J$-limit is already one of the relations used in the $I$-limit.

For a compatible tuple $(x_j{)}_{j\in J}$ and $i\in I$, let $A_i:=\{{\phi }_{ij}(x_j):j\in J,i\le j\}\mathrm{.}$ Cofinality makes $A_i$ nonempty. If $j,j^{\mathrm{\prime }}$ both dominate $i$, choose $k\in J$ dominating both; compatibility gives ${\phi }_{ij}(x_j)={\phi }_{ik}(x_k)={\phi }_{i{j}^{\mathrm{\prime }}}(x_{j^{\mathrm{\prime }}})\mathrm{.}$ Thus $A_i$ is a singleton. Define $\sigma ((x_j){)}_i$ to be its unique member. This makes no simultaneous choice, and the transition identities show that the resulting tuple is compatible on $I$.

By construction, $\rho \circ \sigma$ and $\sigma \circ \rho$ fix every coordinate, hence are identity maps on the corresponding inverse limits. Therefore $\rho$ and $\sigma$ are inverse isomorphisms.