Monomorphisms of Coalgebras

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, $ϕ: C \to D$ is a monomorphism of coalgebras if and only if the first cohomology groups of the coalgebras $C$ and $D$ coincide if and only if $\sum_{i \in I}ε(a^{i})b^{i} = \sum_{i \in I} a^{i} ε(b^{i})$, 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.