= Solution
Give each nonempty <finite set> $X_j$ the <discrete topology>. The <product space> $X=\prod_jX_j$ is <compact> by the <Tychonoff theorem>. For each $i\leq j$, the compatibility condition $f_{ij}(x_j)=x_i$ defines a <closed subset> $C_{ij}\subseteq X$.
These sets have the <finite intersection property>. Indeed, for finitely many conditions choose an index $k$ above every index occurring in them, choose any $x_k\in X_k$, and use the transition maps from $k$ to define all required coordinates; choose the remaining coordinates arbitrarily. Compactness therefore gives
$$
\varprojlim_jX_j=\bigcap_{i\leq j}C_{ij}\ne\varnothing.
$$
This is the <nonemptiness theorem for inverse limits of finite sets>.
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