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How to prove this?

The forgetful functor $U:\mathbf{Top}\to\mathbf{Set}$ has a right adjoint, namely the functor $\mathbf{Set}\to\mathbf{Top}$ which equips a set with the indiscrete topology and left adjoint which equips a set with the discrete topology.

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If $X$ is a topological space and $S$ a set, it should be quite clear that a map from $U(X)$ to $S$ is "the same" as a continuous map from $X$ to $S$ equipped with indiscrete topology, and that a map from $S$ to $U(X)$ is "the same" as a continuous map from $S$ with discrete topology to $X$.

Note that this uses essentially that all maps from a discrete or to an indiscrete space are continuous.

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    $\begingroup$ Can it be generalized for any concrete category with an initial and a terminal object? $\endgroup$
    – porton
    Oct 26, 2013 at 15:43
  • $\begingroup$ Are the units of adjunction identity morphisms ? $\endgroup$ Jun 12, 2022 at 17:24

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