\mathbb A $, whose universe is $ \mathbb A = \setnew{u\in \prod_{n\in {\mathbb N}}A_n}{\forall n\in {\mathbb N}\ u(n)= {\varphi}_n^{n+1}(u(n+1))} $ and such that $r^{ \mathbb A }(u_1, \ldots , u_j)\Leftrightarrow\forall n\in {\mathbb N}\ r^{A_n}(u_1(n), \ldots , u_j(n))$, for every $j$-ary relation symbol $r\in \mathcal L $. We denote by $ {\varphi}_n: \mathbb A \to A_n$ the $n$-th projection map: this is an epimorphism.

A *fundamental sequence* for $\mathcal G$ is a projective sequence $(A_n, {\varphi}_n^m)$ such that the following properties hold:

-   $\{ A_n\}_{n\in {\mathbb N}}$ is cofinal in $\mathcal G$;

-   for any $n$, any $A, B\in \mathcal G $ and any epimorphisms $ \theta_1 :B\to A$, $ \theta_2:A_n\to A$, there exist $m\geq n$ and an epimorphism $\psi :A_m\to B$ such that $ \theta_1\psi= \theta_2 {\varphi}_n^m$.

To study projective Fraïssé limits it is enough to consider fundamental sequences, due to the following fact whose details can be found in [@Camerl2010].

\[pFfpFlfs\] Let $ \mathcal G $ be a nonempty, at most countable (up to isomorphism) family of finite $ \mathcal L $-structures. Then the following are equivalent.

1.  $ \mathcal G $ is a projective Fraïssé family;

2.  $ \mathcal G $ has a projective Fraïssé limit;

3.  $ 