fundamental isomorphism theorem for coalgebras

Let $(C,\mathrm{\Delta },\epsilon )$ and $(D,{\mathrm{\Delta }}^{\prime },{\epsilon }^{\prime })$ be coalgebras. Recall, that if $D_0\subseteq D$ is a subcoalgebra, then $(D_0,{\mathrm{\Delta }}_{|D_0}^{\prime },{\epsilon }_{|D_0}^{\prime })$ is a coalgebra. On the other hand, if $I\subseteq C$ is a coideal, then there is a canonical coalgebra structure on $C/I$ (please, see this entry (http://planetmath.org/SubcoalgebrasAndCoideals) for more details).

Theorem. If $f:C\to D$ is a coalgebra homomorphism, then $\ker (f)$ is a coideal, $\mathrm{im}(f)$ is a subcoalgebra and a mapping $f^{\prime }:C/\ker (f)\to \mathrm{im}(f)$ defined by $f^{\prime }\left(c+\ker (f)\right)=f(c)$ is a well defined coalgebra isomorphism.