  The current work is motivated by the papers $[B_3]$, $[B_6]$, $[Be]$,
$[Be-Tu]$. In particular, using Theorem 3.7 of $[B_3]$ and methods developed in
this paper, the spectral and strong homology groups of continuous maps were
defined and studied $[B_6]$, $[Be]$, $[Be-Tu]$. In this paper we will show that
strong homology groups of continuous maps are a homology type functor, which is
a strong shape invariant and has the semi-continuous property. We will
formulate the new axioms and the conjunction on the uniqueness of the
constructed functor.
