\mathcal G $ has a fundamental sequence.

If these conditions hold and $ \mathcal G_0$ is cofinal in $ \mathcal G $ then $\mathcal G_{0}$ is a projective Fraïssé family and the projective Fraïssé limits of $\mathcal G _{0}, \mathcal G $, and of the fundamental sequence coincide. A projective Fraïssé limit for them is the projective limit of the fundamental sequence.

If $\mathcal G$ is a projective Fraïssé family, one can check whether a given projective sequence is fundamental for $\mathcal G$ with the following.

\[provefundseq\] Let $\mathcal G$ be a projective Fraïssé family of topological ${\mathcal{L}}$-structures. Let $(A_{n}, {\varphi}_{n}^m)$ be a projective sequence in $\mathcal G$. Assume that for each $A\in \mathcal G $, $n \in {\mathbb N}$, and epimorphism $\theta:A \to A_{n}$, there exist $m\ge n$ and an epimorphism $\psi: A_{m} \to A$ such that $\theta\psi= {\varphi}^{m}_{n}$. Then $(A_n, {\varphi}_n^m)$ is a fundamental sequence for $ \mathcal G $.

(F1) Let $A\in \mathcal G $, by (JPP) there exist $A' \in \mathcal G$, and epimorphisms ${\varphi}: A' \to A$ and ${\varphi}': A' \to A_{0}$. By hypothesis there are $n$ and an epimorphism $\theta:A_{n} \to A'$ such that ${\varphi}' \theta = {\varphi}^{n}_{0}$. Then ${\varphi}\theta$ is an epimorphism $A_{n} 