Subcoalgebras, coideals, and quotient coalgebras

Let $(C,{\mathrm{\Delta }}_C,{\epsilon }_C)$ be a coalgebra. For every subspace $U\subseteq C$, write $j_U:U\hookrightarrow C$ for its inclusion and interpret the image of $U\otimes C$ or $C\otimes U$ in $C\otimes C$ through the tensor maps induced by inclusion.

A subcoalgebra of $C$ is a subspace $D\subseteq C$ equipped with linear maps ${\mathrm{\Delta }}_D:D\to D\otimes D$ and ${\epsilon }_D:D\to k$ satisfying the coalgebra axioms, such that the inclusion $j_D$ is a coalgebra map. Under AC, Tensoring injections and the kernel of a tensor product of quotient maps over a field makes $j_D\otimes j_D$ and its iterated tensor powers injective. Hence the equivalent image criterion is that ${\mathrm{\Delta }}_C(D)$ lie in the image of $D\otimes D$ and that the counit on $D$ be the restriction of ${\epsilon }_C$: the inverse of $j_D\otimes j_D$ on its image gives the unique linear ${\mathrm{\Delta }}_D$, and composing its axioms with the injective tensor powers reduces them to the coalgebra axioms on $C$.

A coideal is a subspace $I\subseteq C$ satisfying ${\epsilon }_C(I)=0,{\mathrm{\Delta }}_C(I)\subseteq \mathrm{im}(I\otimes C\to C\otimes C)+\mathrm{im}(C\otimes I\to C\otimes C)\mathrm{.}$ Under AC the two tensor maps are injective, so the second condition is written in the standard shorthand ${\mathrm{\Delta }}_C(I)\subseteq I\otimes C+C\otimes I$. Both conditions are part of the definition.

A quotient coalgebra by $I$ means a coalgebra structure on the vector-space quotient $C\mathrm{/}I$ for which the canonical projection $q:C\to C\mathrm{/}I$ is a coalgebra map. The formula for that structure is not part of this definition.