Let $k$ be a commutative ring, which I am unwilling to assume is a field, and suppose $(C_i)$ is a collection of coassociative $k$-subcoalgebras1 of a coassociative $k$-coalgebra $C$. Is there always a coalgebra structure on the $k$-submodule $\bigcap C_i$ making it a subcoalgebra of $C$?
If not, is there some reasonable hypothesis (cocompleteness, grading, ...) on $C$ or the submodules $C_i$ guaranteeing it?
In the case $k$ is indeed a field, there is the standard proof summing annihilator ideals in the dual algebra $C^*$, but that technique doesn't seem to be available in the general case.
1 This means, each $C_i$ is a both a coalgebra in some way and a $k$-submodule of $C$ such that $C_i \hookrightarrow C$ is a coalgebra map, the issue with a less finicky definition being that $C_i \otimes_k C_i \to C \otimes_k C$ isn't in general an injection.
