Unbounded sets of maps and compactification in extension theory

Suppose that K is a CW-complex. When we say that a space Y is an absolute co-extensor for K, we mean that K is an absolute extensor for Y, i.e., that for every closed subset A of Y and any map , there exists a map that extends f.Our main theorem will provide several statements that are equivalent to the condition that whenever K is a CW-complex and X is a space which is the topological sum of a countable collection of compact metrizable spaces each of which is an absolute co-extensor for K, then the Stone-Čech compactification of X is an absolute co-extensor for K.