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Let $FP(X)$ be the free paratopological group over a topological space $X$.For each non-negative integer $n\in\mathbb{N}$, denote by $FP_{n}(X)$ thesubset of $FP(X)$ consisting of all words of reduced length at most $n$, and$i_{n}$ by the natural mapping from $(X\bigoplus X^{-1}\bigoplus\{e\})^{n}$ to$FP_{n}(X)$. In this paper, we mainly improve some results of A.S. Elfard andP. Nickolas's [On the topology of free paratopological groups. II, TopologyAppl., 160(2013), 220--229.]. The main result is that the natural mapping$i_{2}: (X\bigoplus X_{d}^{-1}\bigoplus\{e\})^{2}\longrightarrow FP_{2}(X)$ isa closed mapping if and only if every neighborhood $U$ of the diagonal$\Delta_{1}$ in $X_{d}\times X$ is a member of the finest quasi-uniformity on$X$, where $X$ is a $T_{1}$-space and $X_{d}$ denotes $X$ when equipped withthe discrete topology in place of its given topology.

The mapping $i_{2}$ on the free paratopological groups