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](http://plms.oxfordjournals.org/content/s3-40/2/298.full.pdf), 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?



Is there a left-adjoint to the restriction of comodules?
If we only consider coalgebras over a fixed ground *field*, then Takeuchi's article [Morita theorems for categories of comodules](http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/6208/1/jfs240306.pdf) 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.


