The main result of this note is the following: Theorem A. If
f
:
X
→
Y
f:X \to Y
is an epimorphism of
H
C
W
∗
\mathcal {H}\mathcal {C}{\mathcal {W}^*}
, the homotopy category of pointed path-connected CW-spaces, and
π
1
(
f
)
:
π
1
(
X
)
→
π
1
(
Y
)
{\pi _1}(f):{\pi _1}(X) \to {\pi _1}(Y)
is a monomorphism, then
f
~
:
X
~
→
Y
~
\tilde f:\tilde X \to \tilde Y
is an epimorphism of
H
C
W
∗
\mathcal {H}\mathcal {C}{\mathcal {W}^*}
. As a straightforward consequence the following results of Dyer-Roitberg (Topology Appl. (to appear)) is derived: Theorem B. A map
f
:
X
→
Y
f:X \to Y
is an equivalence in
H
C
W
∗
\mathcal {H}\mathcal {C}{\mathcal {W}^*}
, the homotopy category of pointed path-connected CW-spaces, iff it is both an epimorphism and a monomorphism in
H
C
W
∗
\mathcal {H}\mathcal {C}{\mathcal {W}^*}
.