  Let $\mathcal{A}$ and $\mathcal{B}$ be monoidal categories and let
$R:\mathcal{A} \rightarrow \mathcal{B}$ be a lax monoidal functor. If $R$ has a
left adjoint $L$, it is well-known that the two adjoints induce functors
$\overline{R}={\sf Alg}(R):{\sf Alg}(\mathcal{A})\rightarrow {\sf Alg
}(\mathcal{B})$ and $\underline{L}={\sf Coalg(L)}:{\sf
Coalg}(\mathcal{B})\rightarrow {\sf Coalg}(\mathcal{A})$ respectively. The pair
$(L,R)$ is called "liftable" if the functor $\overline{R}$ has a left adjoint
and if the functor $\underline{L}$ has a right adjoint. A pleasing fact is
that, when $\mathcal{A}$, $\mathcal{B}$ and $R$ are moreover braided, a
liftable pair of functors as above gives rise to an adjunction at the level of
bialgebras.
  In this note, sufficient conditions on the category $\mathcal{A}$ for
$\overline{R}$ to possess a left adjoint, are given. Natively these conditions
involve the existence of suitable colimits that we interpret as objects which
are simultaneously initial in four distinguished categories (among which the
category of epi-induced objects), allowing for an explicit construction of
$\overline{L}$, under the appropriate hypotheses. This is achieved by
introducing a relative version of the notion of weakly coreflective
subcategory, which turns out to be a useful tool to compare the initial objects
in the involved categories.
  We apply our results to obtain an analogue of Sweedler's finite dual for the
category of vector spaces graded by an abelian group $G$ endowed with a
bicharacter. When the bicharacter on $G$ is skew-symmetric, a lifted adjunction
as mentioned above is explicitly described, inducing an auto-adjunction on the
category of bialgebras "colored" by $G$.
