Let $k$ be a commutative ring and $f: C \to C'$ be a homomorphism of $k$-coalgebras (for simplicity, we can suppose that it is surjective so that $f(C) = C'$). There is a functor: $$f_*: {}^lC-comod \to {}^lC'-comod$$ $$(M, \Delta_M) \mapsto (M, \Delta_M' := (f \otimes id_M) \circ \Delta_M)$$ wherein $\Delta_M$ denotes the left-$C$-coaction on $M$. Does this functor admit left/right-adjoints, and if so, do these left/right-adjoints have concrete descriptions ?
Question edited according to comments by
Mariano Suárez-Álvarez