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If $f\colon C \to D$ is a homomorphism of coalgebras and $\rho\colon V \to V\otimes C$ is a $C$-comodule, then $(1\otimes f)\rho\colon V \to V\otimes D$ is the comodule restricted to $D$. In e.g. Stephen Donkin's article Hopf complements and injective comodules for algebraic groups, section 3, he shows there is a right adjoint to the restriction. I am wondering if there is also a left-adjoint to the restriction?

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Doesn't taking the cotensor product with C over D suffice? – user43326 Jun 25 '14 at 16:50
If I am not mistaken, that gives the right adjoint. – Peter Patzt Jun 25 '14 at 16:52
I believe the answer is no. The corestriction of scalars functor $f_* : \mathrm{Comod}(C) \to \mathrm{Comod}(D)$ does preserve finite limits (they are just limits of the underlying vector spaces). However, infinite products of comodules are not just products of the underlying vector spaces so there is no reason for $f^*$ to preserve them. – Adeel Khan Jun 25 '14 at 22:03
Are we over a field? – Tilman Jul 27 '14 at 18:15
Yes everything is over a field. I am not interested in other cases right now. – Peter Patzt Jul 27 '14 at 18:19

If we only consider coalgebras over a fixed ground field, then Takeuchi's article Morita theorems for categories of comodules seem to prove the existence of left-adjoints of the restriction.

In 1.4 he defines cohomorphisms of $C$-comodules $\operatorname{Cohom}_C(V,V')$ for quasi-finite $V$, that is $\operatorname{Hom}_C(W,V)$ is finite dimensional for every finite dimensional $C$-comodule $W$. He then proves in 1.10 that if $V$ is a $(D,C)$-bicomodule, $\operatorname{Cohom}_C(V,-)$ is the left-adjoint to the cotensor product $-\square^D V$.

We figure that if $C$ is a quasi-finite $D$-comodule, we have with $\operatorname{Cohom}_D(C,-)$ the left-adjoint to $-\square^C C$, i.e. $$\operatorname{Hom}_C(\operatorname{Cohom}_D(C,V),W) \cong \operatorname{Hom}_D(V,W\square^CC)\cong \operatorname{Hom}_D(V,\operatorname{Res} W).$$

The only question that remains is, for $f\colon C\to D$ when is $C$ a quasi-finite $D$-comodule. But that can be left to the comments or to a new question if urgent.

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