= Adjoint chain for the objects of a category
{title2=$\pi_0\dashv D\dashv O\dashv I$}
For small categories, the object functor $O$ has left adjoint the <discrete category> functor and right adjoint the <indiscrete category> functor. Connected components, defined by zigzags of morphisms, give a left adjoint $\pi_0$ to the discrete functor. There is no further left adjoint: $\pi_0$ fails the equalizer of two functors selecting distinct objects of the indiscrete two-object category. There is no further right adjoint: the indiscrete functor fails the coproduct of two singleton sets.
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