The kernel of a morphism of co-rings is a co-ideal

Question

I would like to show that

The Kernel of a coring morphism $\phi:C\rightarrow C'$ between two $R$-corings $(C,\Delta,\epsilon)$ and $(C',\Delta',\epsilon')$ is a coideal.

The only point that I can't show is that $$\Delta(\text{Ker}(\phi))\subset \text{Ker}(\pi\otimes\pi)$$

This should be straightforward but I can't show it.

EDIT:

$\pi$ is the natural projection of $C$ unto $C/\text{Ker}(\phi)$

The definition I use are

A coring morphism from an $R$-coring $(C,\Delta,\epsilon)$ to another $R$-coring $(C',\Delta',\epsilon')$ is a $R$-bimodule morphism $\phi$ such that $\epsilon'\circ\phi = \epsilon$ and $(\phi\otimes\phi)\circ\Delta = \Delta'\circ\phi$.

A coideal $I$ of the $R$-coring $C$ is a $R$-subbimodule of $C$ such that $I\subset \text{Ker}(\epsilon)$ and \begin{align*} \Delta(I) \subset \text{Ker}(\pi\otimes\pi) \end{align*} where $\pi:C\rightarrow C/I$ is the cannonical projection map.

Answer 1

The statement:

The Kernel of a coring morphism $\phi:C\rightarrow C'$ between two $R$-coring's $(C,\Delta,\epsilon)$ and $(C',\Delta',\epsilon')$ is a coideal.

is not true in general:

• It is true if one considers the special case of coalgebras over a field $k$ (a proof of this, can be found at Hopf algebras: an introduction, Dascalescu et al, p. 25-26, proposition 1.4.9),
• However, in the general case that we have an $A$-coring $C$ instead of a coalgebra over a field $k$ (now the linear structure of $C$ is that of an $A$-bimodule ${}_AC_A$, where $A$ is an arbitrary -that is: non-commutative in general- algebra) the statement is not valid as it is: In this case, the coideal is by definition the kernel of a surjective $A$-coring morphism (and not of any $A$-coring morphism). For this last case, see the proof and futher details at: Corings and Comodules, Brzezinski et al, proposition 17.14, p. 177-178.
  The same situation holds in the case of coalgebras over a commutative ring $R$ (for a proof of the case where the "scalars" are elements of a commutative ring $R$, see: Corings and Comodules, Brzezinski et al., proposition 2.4, p. 9-10)