  We prove new necessary and sufficient conditions for a morphism of coalgebras
to be a monomorphism, different from the ones already available in the
literature. More precisely, $\phi: C \to D$ is a monomorphism of coalgebras if
and only if the first cohomology groups of the coalgebras $C$ and $D$ coincide
if and only if $\sum_{i \in I}\epsilon(a^{i})b^{i} = \sum_{i \in I} a^{i}
\epsilon(b^{i})$, for all $\sum_{i \in I}a^{i} \otimes b^{i} \in C \square_{D}
C$. In particular, necessary and sufficient conditions for a Hopf algebra map
to be a monomorphism are given.
