Functor properties of the $\Omega$-saturation of a topological space

Herein, we consider the $\Omega$-saturations of a topological space $X$, which are canonically embedded in the Wallman extension $\omega X$ and are a weakening of the concept of the countably-compactification in the Morita sense. We find necessary and sufficient conditions of the continious extension of a map $X \xrightarrow{f} Y$ to $\Omega$-saturations of the spaces $X$ and $Y$, as well as sufficiently wide categories on which the covariant functors arising in this case are defined.