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Let $H$ be a Hopf algebra, $(V,\Delta_R)$ a right $H$-comodule map, and $f:V \to V$ a right $H$-comodule map. Since by definition we must have, for all $v \in V$, that $$ \Delta_R(f(v)) = \sum f(v_{(0)}) \otimes v_{(1)}, $$ it seems to me that the kernel of $f$ must be a right $H$-sub-comodule. Since for any non-zero $v \in \ker(f)$, $$ 0 = \Delta_R(0) = \Delta_R(f(v)) = \sum f(v_{(0)}) \otimes v_{(1)}. $$ Does this all seem ok?

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