The conditions for the convergence of isotonic operators in partially ordered sets with convergence classes

The necessary and sufficient conditions are given which must be satisfied by an element $x$ of a partially ordered set $X$ with convergence close to topological in order that every net of isotonic operators of the set $X'\subset X$ in $X$, converging on some set $G\subset X'$ to the identity operator, should converge on $X$ to $x$.