= Solution
Let $C\mathcal A=\pi_0\mathcal A$ be the <set> of <connected components of a category>: two <objects of a category> are identified when joined by a finite zigzag of <morphisms>. Let $DA$ be the <discrete category> on $A$, and let $IA$ be the <indiscrete category> on $A$, with exactly one <morphism> for every ordered pair of objects.
A <functor> $\mathcal A\to DA$ must give the same object value along every arrow, hence along every zigzag. Conversely a function $\pi_0\mathcal A\to A$ uniquely defines such a <functor>. A <functor> $DA\to\mathcal A$ is just a choice of an <object of a category> for each element of $A$. A <functor> $\mathcal A\to IA$ is likewise determined by an arbitrary object function, since its arrow images are forced and always compose correctly. Therefore
$$
\operatorname{Cat}(\mathcal A,DA)\cong\mathbf{Set}(C\mathcal A,A),\qquad
\operatorname{Cat}(DA,\mathcal A)\cong\mathbf{Set}(A,O\mathcal A),
$$
$$
\operatorname{Cat}(\mathcal A,IA)\cong\mathbf{Set}(O\mathcal A,A).
$$
This is the <adjoint chain for the objects of a category>:
$$
\boxed{C\dashv D\dashv O\dashv I.}
$$
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