An eventually constant inverse system has inverse limit equal to its stable value

A compatible tuple is determined uniquely by its $i_0$-coordinate, because for every $j\ge i_0$ the coordinate $g_j$ must be the unique preimage of $g_{i_0}$ under the isomorphism ${\phi }_{i_{0}j}$.

Conversely, let $g\in G_{i_0}$. For each index $i$, choose some $j\ge i,i_0$, possible because the index set is directed, and define $g_i:={\phi }_{ij}({\phi }_{i_{0}j}^{-1}(g))\in G_i\mathrm{.}$ This does not depend on the choice of $j$, because any larger common upper bound gives the same value after applying compatibility of the transition maps. The tuple $(g_i)$ is compatible and has $i_0$-coordinate $g$. Therefore the projection to the $i_0$-coordinate is surjective as well as injective by step 1.1.

The map sending a compatible tuple to its $i_0$-coordinate is therefore a bijective homomorphism, and [L1] identifies it with the inverse-limit comparison map. Hence the inverse limit is isomorphic to $G_{i_0}$.