  This paper is a continuation of [BaSh], where we studied the behaviour of the
abelianization functor under inverse limits. Our main result in [BaSh] was that
if $\mathcal{T}$ is a countable directed poset and $G:\mathcal{T}\to\mathcal{G}
rp$ is a diagram of groups that satisfies the Mittag-Leffler condition, then
the natural map $$\mathrm{Ab}({\lim}_{t\in\mathcal{T}}G_t)\to
{\lim}_{t\in\mathcal{T}}\mathrm{Ab}(G_t)$$ is surjective, and its kernel is a
cotorsion group. The abelianization is an example of a left adjoint functor
from groups to abelian groups.
  In this paper we study the behaviour under inverse limits of left adjoint
functors from pointed sets to abelian groups. Such functors are classified by
abelian groups, where to the abelian group $A$ corresponds the left adjoint
functor $L_A:\mathcal{S} \text{et}_*\to\mathcal{A} \text{b}$ given by
$L_A(Y)=\bigoplus_{Y\setminus\{*\}}A.$ If $\mathcal{T}$ is a directed poset and
$X:\mathcal{T}\to\mathcal{S} \text{et}_*$ is a is diagram of pointed sets, we
show that the natural map $$\rho:L_A({\lim}_{t\in\mathcal{T}}X_t)
\to{\lim}_{t\in\mathcal{T}}L_A(X_t)$$ is injective. If, in addition,
$\mathcal{T}$ is countable and $X$ satisfies the Mittag-Leffler condition, we
show that the cokernel of $\rho$ is an algebraically compact group. Compared
with the main result in [BaSh], algebraically compact is much stronger then
cotorsion as it also requires the Ulm length to be $\leq 1$.
  We also show that this result, even in its weak form of cotorsion, does not
extend to uncountable diagrams. Namely, if $A$ is not the product of a
divisible group and a bounded group, we construct a directed poset
$\mathcal{T}$ with $|\mathcal{T}|=2^{\aleph_0}$ and a diagram
$X:\mathcal{T}\to\mathcal{S} \text{et}_*$, that satisfies the Mittag-Leffler
condition, such that the cokernel of $\rho$ is not cotorsion.
