Let ${(e_{\beta }:𝐐\to Y_{\beta })}_{\beta \in \mathbf{\text{Ord}}}$ be the large source of epimorphisms in the category $\mathbf{\text{Ury}}$ of Urysohn spaces constructed in [2]. A sink ${(g_{\beta }:Y_{\beta }\to X)}_{\beta \in \mathbf{\text{Ord}}}$ is called natural, if $g_{\beta }\circ e_{\beta }=g_{{\beta }^{\text{'}}}\circ e_{{\beta }^{\text{'}}}$ for all $\beta ,{\beta }^{\text{'}}\in \mathbf{\text{Ord}}$. In this paper natural sinks are characterized. As a result it is shown that $\mathbf{\text{Ury}}$ permits no $(Epi,ℳ)$-factorization structure for arbitrary (large) sources.

In 1985, V. G. Pestov described a neighborhood base at the identity of free topological groups on a Tychonoff space in terms of the elements of the fine uniformity on the Tychonoff space. In this paper, we extend Postev’s description to the free paratopological groups where we introduce a neighborhood base at the identity of free paratopological groups on any topological space in terms of the elements of the fine quasiuniformity on the space.