Let $(C,\Delta,\varepsilon)$ be a coalgebra over a field $k$

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

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

One can show that if $D\subseteq C$ is a subcoalgebra, then $(D, \Delta_{| D}, \varepsilon_{| D})$ 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 $\Delta(x)=\sum a_i\otimes b_i$ then we define $$\Delta':C/I\to (C/I)\otimes (C/I);$$ $$\Delta'(x+I)=\sum (a_i+I)\otimes (b_i+I)$$ and $\varepsilon':C/I\to k$ as $\varepsilon'(x+I)=\varepsilon(x)$ One can show that these two maps are well defined and $(C/I,\Delta',\varepsilon')$ is a coalgebra.