I think this question Compact Hausdorff and C^*-algebra "objects" in a category. shows that there is no reciprocity between categories of algebra-objects of two algebraic categories.
So, what I mean here is that if $\mathcal{A}$ and $\mathcal{B}$ are two algebraic categories, define categories $\mathcal{A}(\mathcal{B})$ of $\mathcal{A}$-objects in $\mathcal{B}$ and $\mathcal{B}(\mathcal{A})$ of $\mathcal{B}$-objects of $\mathcal{A}$. For example if $\mathcal{A}$ is a category of groups $\mathsf{GRP}$, and $\mathcal{B}$ is a category of compact Hausdorff spaces $\mathsf{KH}$. Then $\mathsf{GRP}(\mathsf{KH})$ are just topological groups with compact Hausdorff topology, which sounds pretty reasonable. On the other hand objecs of $\mathsf{KH(GRP)}$ are groups but the "compact Hausdorff topology" on them may not be a conventional topology but something else instead.
So, what can we say about relations between $\mathcal{A(B)}$ and $\mathcal{B(A)}$ in general? Are they equivalent, or isomorphic, or neither? Are there any conditions on categories $\mathcal{A}$ and $\mathcal{B}$ which ensures some Reciprocity? For example, if $\mathcal{A}$ and $\mathcal{B}$ are finitary algebraic theories.
Sorry, I am not very well versed algebraic categories and Monads. But I have red a chapter in MacLane's CFWM, and a paragraph on Peter Johnstone's "Stone Spaces". I still feel that, if any such results exist, they might be well known in the field. I also have seen an article "Compact Hausdorff objects" by E.G. Manes. It contains some explicit constructions of form $\mathsf{KH(GRP)}$. But I'm not sure if it is relevant, as Manes also considers categories equipped with a closure operator, and it may be different from "monadic" $\mathsf{KH(GRP)}$. In Manes's case closed sets in groups are subgroups and every group is Hausdorff compact, so $\mathsf{KH(GRP)} = \mathsf{GRP}$
Addendum:
Aftere thinking a while on what Tom Leinster wrote in his answer https://mathoverflow.net/a/14895/91850 , I now think that $\mathsf{KH(GRP)} \cong \mathsf{GRP(KH)}$.
So, if for any group $G$ in $\mathsf{KH(GRP)}$ and for any ultrafilter $U$ on any set $I$ there are group homomorphisms $\xi_U : G^I \to G$, which represensts taking a limit over ultrafilter $U$, this $\xi_U$ should define a compact Hausdorff topology on $G$. As each $\xi_U$ is a homomorphism this topology mus be a group topology. This should define an isomorphism of categories.
