If $T=\ObjTerm{\ECat}$, we obtain a natural bijection
$$
\Hom{\ECat}{\ObjTerm{\ECat}}{\ObjMono{X}{Y}} \cong \Monos{\ECat}{X}{Y}
$$
Thus the points of $\ObjMono{X}{Y}$ correspond exactly to the monos from $X$ to $Y$ in $\ECat$. But these do not entirely determine the object $\ObjMono{X}{Y}$.