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Notice that $F(V)=V^H$ for all $V\in\Com$.

If $$0\to \tag{\star}0\to W\to V\to U\to0$$ is a short exact sequence in $\Com$, then we have a long exact sequence for the derived functors $R^pF$, which starts with $$0\to W^H\to V^H\to U^H\to\Ext^1_{\Com}(k,W)\to\cdots$$

We can conclude, then, as usual, that the map $V^H\to U^H$ is surjective if, for example $\Ext^1_{\Com}(k,W)=0$. This can happen for various reasons: one obvious one is that $W$ be an injective comodule. A draconiant version of this is the condition that $H$ be cosemisimple.

To say something more intelligent, one would probably need to know more details about your concrete situation, though.


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Notice that $F(V)=V^H$ for all $V\in\Com$.

If $$0\to W\to V\to U\to0$$ is a short exact sequence in $\Com$, then we have a long exact sequence for the derived functors $R^pF$, which starts with $$0\to W^H\to V^H\to U^H\to\Ext^1_{\Com}(k,W)\to\cdots$$

We can conclude, then, as usual, that the map $V^H\to U^H$ is surjective if, for example $\Ext^1_{\Com}(k,W)=0$. This can happen for various reasons: one obvious one is that $W$ be an injective comodule. A draconiant version of this is the condition that $H$ be cosemisimple.

To say something more intelligent, one would probably need to know more details about your concrete situation, though.