  A problem of constructing of local definitions for formations of finite
groups is discussed in the article. The author analyzes relations between local
definitions of various types. A new proof of existence of an
$\omega$-composition satellite of an $\omega$-solubly saturated formation is
obtained. It is proved that if a non-empty formation of finite groups is
$\mathfrak X$-local by F\"{o}rster, then it has an $\mathfrak X$-composition
satellite.
