Consider the following quote from the Wikipedia entry Coalgebra:

The kernel of every coalgebra morphism $f : C_1 \to C_2$ is a coideal in $C_1$, and the image is a subcoalgebra of $C_2$.

I can't see any qualifiers preceding or succeeding the statement. Am I missing something obvious here, or is this just plain wrong?

Surely it should say

The kernel of a coalgebra morphism $f : C_1 \to C_2$ is a coideal in $C_1$ if, and only if, the image is a subcoalgebra of $C_2$.

Consider the following quote from the Wikipedia entry Coalgebra:

The kernel of every coalgebra morphism $f : C_1 \to C_2$ is a coideal in $C_1$, and the image is a subcoalgebra of $C_2$.

I can't see any qualifiers preceding or succeeding the statement. Am I missing something obvious here, or is this just plain wrong?

Do there not exists kernels of coalgebra maps that are not coideals?