Fraïssé family* if the following properties hold:

1.  (joint projection property) for every $A, B\in \mathcal G $ there are $C\in \mathcal G $ and epimorphisms $C\to A$, $C\to B$;

2.  (amalgamation property) for every $A, B, C\in \mathcal G $ and epimorphisms $ {\varphi}_1:B\to A$, $ {\varphi}_2:C\to A$ there are $D\in \mathcal G $ and epimorphisms $\psi_1:D\to B$, $\psi_2:D\to C$ such that $ {\varphi}_1\psi_1= {\varphi}_2\psi_2$.

Given a family $ \mathcal G $ of topological $ \mathcal L $-structures, a topological $ \mathcal L $-structure $ \mathbb L $ is a *projective Fraïssé limit* of $ \mathcal G $ if the following hold:

1.  \[Lone\] (projective universality) for every $A\in \mathcal G $ there is some epimorphism $ \mathbb L \to A$;

2.  for any clopen covering $ \mathcal U $ of $ \mathbb L $ there are $A\in \mathcal G $ and an epimorphism $ \mathbb L \to A$ refining $ \mathcal U $.

3.  \[Lthree\] (projective ultrahomogeneity) for every $A\in \mathcal G $ and epimorphisms $ {\varphi}_1, {\varphi}_2: \mathbb L \to A$ there exists an automorphism $\psi \in \operatorname{Aut}(\mathbb L )$ such that $ {\varphi}_2= {\varphi}_1\psi $.

Note that in the original definition of a projective Fraïssé limit 