subcoalgebras and coideals

Let $(C,\mathrm{\Delta },\epsilon )$ be a coalgebra over a field $k$.

Definition. Vector subspace $D\subseteq C$ is called subcoalgebra iff $\mathrm{\Delta }(D)\subseteq D\otimes D$.

Definition. Vector subspace $I\subseteq C$ is is called coideal iff $\mathrm{\Delta }(I)\subseteq I\otimes C+C\otimes I$ and $\epsilon (I)=0$.

One can show that if $D\subseteq C$ is a subcoalgebra, then $(D,{\mathrm{\Delta }}_{|D},{\epsilon }_{|D})$ is also a coalgebra. On the other hand, if $I\subseteq C$ is a coideal, then we can cannoicaly introduce a coalgebra structure on the quotient space $C/I$. More precisely, if $x\in C$ and $\mathrm{\Delta }(x)=\sum a_i\otimes b_i$, then we define

$${\mathrm{\Delta }}^{\prime }:C/I\to (C/I)\otimes (C/I);$$

$${\mathrm{\Delta }}^{\prime }(x+I)=\sum (a_i+I)\otimes (b_i+I)$$

and ${\epsilon }^{\prime }:C/I\to k$ as ${\epsilon }^{\prime }(x+I)=\epsilon (x)$. One can show that these two maps are well defined and $(C/I,{\mathrm{\Delta }}^{\prime },{\epsilon }^{\prime })$ is a coalgebra.