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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 → 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 ${\sum }_{i\in I}\epsilon \left(a^i\right)b^i={\sum }_{i\in I}{a}^i\epsilon \left(b^i\right)$ for all ${\sum }_{i\in I}{a}^i\otimes b^i\in C{◻}_{D}C$. In particular, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism are given.