Proposition B.5

There exists an adjunction in which the inclusion of compactly generated spaces in universe $u$ into topological spaces in universe $u$ is left adjoint to the functor $X\mapsto kX$, which leaves underlying functions unchanged. Here compact generation is with respect to compact Hausdorff spaces in universe $u$, and $kX$ has the underlying set of $X$, with a subset open exactly when every originally continuous map from such a compact Hausdorff space has open inverse image of that subset. Thus there exist bijections, natural in every compactly generated space $A$ and every topological space $X$, between continuous maps from the underlying topological space of $A$ to $X$ and continuous maps $A\to kX$; existence of an adjunction, without a uniqueness assertion, is claimed.