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 leftadjoint to the restriction?

$\begingroup$ Doesn't taking the cotensor product with C over D suffice? $\endgroup$ – user43326 Jun 25 '14 at 16:50

$\begingroup$ If I am not mistaken, that gives the right adjoint. $\endgroup$ – Peter Patzt Jun 25 '14 at 16:52

$\begingroup$ Are we over a field? $\endgroup$ – Tilman Jul 27 '14 at 18:15

$\begingroup$ Yes everything is over a field. I am not interested in other cases right now. $\endgroup$ – 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 leftadjoints of the restriction.
In 1.4 he defines cohomorphisms of $C$comodules $\operatorname{Cohom}_C(V,V')$ for quasifinite $V$, 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 leftadjoint to the cotensor product $\square^D V$.
We figure that if $C$ is a quasifinite $D$comodule, we have with $\operatorname{Cohom}_D(C,)$ the leftadjoint 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 quasifinite $D$comodule. But that can be left to the comments or to a new question if urgent.