Efimov has in his recent preprint, Appendix F, a long table of analogies between the categories $\text{Cat}^{dual}_{st}$ and $\text{CompHaus}^{op}$.

For example, both categories have a single $\omega_1$-compact generator ($\text{Shv}_{\geq 0}(\mathbb{R})$ and $[0,1]$), and they have coreflective subcategories given by compactly generated categories, $\text{Cat}^{cg}_{st}$, and profinite spaces, respectively.

Is there a way to make this table a bit more precise, say by turning it into a functor that preserves all the claimed properties on both sides? A natural guess would be to consider the functor $\text{CompHaus}^{op} \rightarrow \text{Cat}^{dual}_{st}, X \mapsto \text{Shv}(X; \text{Sp})$. But that for example doesn't map the interval to the $\omega_1$-compact generator. Another point is that maybe inside the larger category of compactly assembled categories this analogy can be made more precise?

Efimov has in his recent preprint K-theory and localizing invariants of large categories, Appendix F, a long table of analogies between the categories $\text{Cat}^\text{dual}_\text{st}$ and $\text{CompHaus}^\text{op}$.

For example, both categories have a single $\omega_1$-compact generator ($\operatorname{Shv}_{\geq 0}(\mathbb{R})$ and $[0,1]$), and they have coreflective subcategories given by compactly generated categories, $\text{Cat}^\text{cg}_\text{st}$, and profinite spaces, respectively.

Is there a way to make this table a bit more precise, say by turning it into a functor that preserves all the claimed properties on both sides? A natural guess would be to consider the functor $\text{CompHaus}^\text{op} \rightarrow \text{Cat}^\text{dual}_\text{st}$, $X \mapsto \operatorname{Shv}(X; \text{Sp})$. But that for example doesn't map the interval to the $\omega_1$-compact generator. Another point is that maybe inside the larger category of compactly assembled categories this analogy can be made more precise?