Let $k$ be a field, $C$ a $k$-coalgebra, and $M$ a left $C$-comodule. Then, for a short exact sequence


$$
0 \rightarrow N \rightarrow M \rightarrow L \rightarrow 0
$$


of vector spaces, we have that $N$ and $L$ are also $C$-comodules. Does the converse hold?


I've tried using the correspondence of left $C$-comodules and rational right $C^\ast$-modules, but it's not clear to me why the map $\varphi: M\rightarrow M \otimes C$ in the definition of rational modules exists in general. It's clear it exists if the short exact sequence splits.



$M$ comodule if and only if $N$ and $L$ comodules
For any (coassociative, counital) coalgebra $C$ over a field $k$, there is a fully faithful exact functor from the category of $C$-comodules to the category of $C^\*$-modules. The essential image of this functor is the full subcategory of rational $C^\*$-modules (hence one can identify $C$-comodules with rational $C^\*$-modules).


The full subcategory of rational $C^\*$-modules is closed under subobjects and quotient objects in the category of $C^\*$-modules. But it is not closed under extensions, generally speaking.


For a counterexample, consider the matrix coalgebra $C=\begin{pmatrix} k & V \\ 0 & k\end{pmatrix}$, where $V$ is a $k$-vector space. Its dual algebra $C^\*$ is the matrix algebra $C^\*=\begin{pmatrix} k & V^\* \\ 0 & k\end{pmatrix}$. Then $C$-comodules $E$ are essentially pairs of vector spaces $(E\_1,E\_2)$ endowed with a linear map $\psi\colon E\_1\longrightarrow V\otimes\_k E\_2$, while $C^\*$-modules are pairs of vector spaces $(M\_1,M\_2)$ endowed with a linear map $g\colon V^\*\otimes\_k M\_1\longrightarrow M\_2$. The functor from $C$-comodules to $C^\*$-modules is constructed in the obvious way.


Now take $V$ to be an infinite-dimensional vector space, and let $g\colon V^\*\longrightarrow k$ be a linear function not arising from any element of $V$ (i.e., $g$ belongs to $V^\*{}^\*\setminus V$). Take $M\_1=M\_2=k$ to be one-dimensional vector spaces. Then $M=(k,k,g)$ is an irrational $C^\*$-module. Still it is an extension of two rational $C^\*$-modules,
$$0\longrightarrow N=(0,k)\longrightarrow M=(k,k)\longrightarrow L=(k,0)\longrightarrow0.$$


