Sequential ends and nonstandard infinite boundaries of coarse spaces

This paper is an addendum to the author's previous paper [#Im20a]. Miller et al. [#MSM10] introduced a functor $\sigma\colon\mathbf{pCoarse}\to\mathbf{Sets}$, where $\mathbf{pCoarse}$ is the category of pointed coarse spaces and coarse maps. DeLyser et al. [#DLT13] introduced a functor $\varepsilon\colon\mathbf{pCoarse}\to\mathbf{Sets}$, and proved that $\varepsilon$ coincides with $\sigma$ on $\mathbf{pMetr}$ (the full subcategory of metrisable spaces). Using techniques of nonstandard analysis, the author in [#Ima20a] provided a functor $\iota\colon\mathscr{C}\subseteq\mathbf{pCoarse}\to\mathbf{Sets}$, where $\mathscr{C}$ is an arbitrary small full subcategory, and a natural transformation $\omega\colon\sigma\restriction\mathscr{C}\Rightarrow\iota$. The surjectivity of $\omega$ has been proved for all proper geodesic metrisable spaces, while the injectivity has remained open. In this note, we first pointed out that $\omega$ is the composition of two natural transformations $\varphi\restriction\mathscr{C}\colon\sigma\restriction\mathscr{C}\Rightarrow\varepsilon\restriction\mathscr{C}$ and $\omega'\colon\varepsilon\restriction\mathscr{C}\Rightarrow\iota$, and then show that $\omega'$ is injective for all spaces in $\mathscr{C}$. As a corollary, $\omega$ is injective for all metrisable spaces in $\mathscr{C}$. This partially answers some of the problems posed in [#Ima20a].