Let $\Delta$ denote the simplicial category and $\textbf{sSet}$ the category $\text{Func}(\Delta^{\text{op}},\textbf{Set})$. If $X$ is a simplicial set, a simplicial subset of $X$ is a simplicial set $Y$ such that $Y_n\subset X_n$ $(n\in\mathbb{N})$ and $X(\alpha)|_{Y_m}=Y(\alpha)$ for all $\alpha:[n]\rightarrow[m]\in\Delta$.
Now the author writes
We can form images of morphisms of simplicial sets, and unions and intersections of simplicial subsets, in the obvious way.
I'm not sure exactly what this means.
Let $u:X\rightarrow Y$ be a morphism of simplicial sets. What would the "image of $u$" have to be? An element of $\textbf{Set}$, e.g. $\bigsqcup_n u([n])(X_n)$?
If $Y,Z$ are simplicial subsets of $X$, then presumably their union should be another simplicial subset of $X$. In that case, is it the functor that send $[n]$ to $Y_n\sqcup Z_n$? But it's not clear how $Y_n\sqcup Z_n$ can be a subset of $X_n$...If we replace $\sqcup$ by $\cup$, it won't be obvious what the image of $\alpha:[n]\rightarrow[m]$ can be.
Finally, what should the intersection of simplicial subsets $Y,Z$ of $X$ be? Again it's a functor from $\Delta^{\text{op}}$ to $\textbf{Set}$. I am inclined to say that it sends $[n]$ to $Y_n\cap Z_n$. Here again I have no idea what the image of $\alpha:[n]\rightarrow[m]$ should be.