  Let $P_n$ denote the undirected path of length $n-1$. The cardinality of the
set of congruence classes induced by the graph homomorphisms from $P_n$ onto
$P_k$ is determined. This settles an open problem of
  Michels and Knauer (Disc. Math., 309\ (2009)\ 5352-5359). Our result is based
on a new proven formula of the number of homomorphisms between paths.
