A compactly generated space is a space $X$ such that $f : X \rightarrow Y$ is continuous if and only if $K \rightarrow X \stackrel{f}{\rightarrow} Y$ is continuous for each compact hausdorff space $K$.

Let $I$ be the category of compact hausdorff topological spaces with continuous maps. I think being compactly generated is equivalent to the canonical map $\epsilon_X : \int_{K \in I} [K, X]_{\text{Top}} \times K \rightarrow X$ being an isomorphism. $\epsilon_X$ is the counit of an adjunction between functors $L : [ I^{op}, \text{Set}]_{\text{Cat}} \rightarrow \text{Top}$ sending $S$ to $\int_{K \in I} S(K) \times K$ and $R : \text{Top} \rightarrow [ I^{op}, \text{Set}]_{\text{Cat}}$ sending a space $X$ to the presheaf $S : I^{op} \rightarrow \text{Set}$ sending $K$ to $[K, X ]_{\text{Top}}$. $L$ is the left Kan extension of the inclusion $\iota : I \rightarrow \text{Top}$ along the yoneda embedding $Y : I \rightarrow [I^{op}, \text{Set}]_{\text{Cat}}$, as in the diagram below:

That $L$ is left adjoint to $R$ is a famous theorem, which was discussed here.

If I am not mistaken, this adjunction is related to Peter Scholze and Dustin Clausen's condensed mathematics. Surprisingly, there is no information lost if we take instead of $I$ the smaller category of profinite sets. There is a fully faithful functor from compactly generated spaces to condensed sets (sheaves on the étale site of a point) that arises in this way.

My question is, has anyone taken this perspective to get a streamlined, categorical proof that the category of compactly generated spaces is cartesian closed? We might start by observing that the category of presheaves on $I$ has all small limits and colimits, $L$ preserves colimits, and $R$ preserves limits. Might this allow us to carry over the cartesian closed-ness of $[I^{op}, \text{Set}]_{\text{Cat}}$ to the category of compactly generated spaces?

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