\to A$, as wished.

(F2) Let $A, B\in \mathcal G $ and epimorphisms $ \theta_1 :B\to A$, $ \theta_2:A_n\to A$. By (AP) there exist $C\in \mathcal G$ and epimorphisms $\rho_{1}:C\to B$ and $\rho_{2}:C\to A_{n}$ such that $\theta_1\rho_{1}=\theta_{2}\rho_2$. By hypothesis, there exist $m\ge n$ and an epimorphism $\psi': A_{m} \to C$ such that $\rho_{2}\psi'= {\varphi}^{m}_{n}$. Then $\psi = \rho_{1}\psi':A_{m} \to B$ is such that $ \theta_{1}\psi= \theta_{2} {\varphi}^{m}_{n}$.

Notice that the converse of holds as well.

Fine projective sequences {#fineprojectivesequences}
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In the sequel, whenever we denote a language with a subscript, like in $ \mathcal L_R$, we mean that the language contains a distinguished binary relation symbol represented in the subscript. The interpretation of $R$ in an ${\mathcal{L}}_R$-topological structure is expected to be reflexive and symmetric. These properties are preserved under projective limits. A *prespace* is any $ {\mathcal{L}}_R$-topological structure $A$ where the interpretation of $R$ is also transitive, that is, an equivalence relation; in this case, we say that $A$ is a prespace of $ {{\raisebox{.1em}{$A\!$}\left/\raisebox{-.1em}{$R^A$}\right.}} $. Since $R^{A}$ is a closed equivalence relation, the quotient map $p:A\to {{\raisebox{.1em}{$A\!$}\left/\raisebox{-.1em}{$R^A$}\right.}} $ is closed. Notice that $ {{\raisebox{.1em}{$A\!$}\left/\raisebox{-.1em}{$R^A$}\right.}} $ is then endowed with an $( {\mathcal{L}}_R\setminus \set{R} )$-structure, where $r^{A/R^A}=p\times\ldots\times p(r^A)$, 