# Compact objects in categories of comodules.

Are the finite dimensional comodules the compact objects in the category of comodules over a Hopf algebra? If yes, is there a reference? If no, which are the compact objects? Here by compact I mean that the Hom functor from the compact object preserves filtered colimits. Thank you!

• I believe this is true over a field, at least, and that it follows from the fact that every comodule is a filtered colimit of finite-dimensional comodules. – Qiaochu Yuan Oct 24 '15 at 18:08
• Thank you! How would you prove this? Is there a natural right adjoint to the Hom functor in this case? – m.pap Oct 24 '15 at 20:58