A modification of the Harris ($WO$) condition and functor properties of the hyperspace and Wallman compactification

A necessary and sufficient condition for the existence of a continuous extension $\text{exp} X \overset{\bar{f}}{\longrightarrow} \text{exp} Y$ of a map $X \overset{f}{\longrightarrow} Y$ is found, where $\text{exp} X$ is a hyperspace of the space $X$ endowed with Vietoris topology, and the map $\bar{f}$ is defined as $\bar{f}(F) = [f(F)]_Y$ ($[ \cdot ]_Y $ is a closure operator on the space $Y$). The obtained condition (named as $(\omega o)$ condition) is a modification of the Harris $(WO)$ condition. It is also shown, that ($\omega o$) condition is sufficient and in the case of regularity of a space $Y$ it is necessary for the existence of multivalued upper semi-continuous extension $\omega X \overset{\tilde{f}}{\longrightarrow} \omega Y$ of a map $f$ which satisfies some additional conditions ($\omega X$ is the Wallman compactification). The results obtained are commented on by category theory.