THIS IS NOT AN ANSWER, rather an additional question

I always wanted to know how the following purely abstract-nonsensical (category-theoretic) constructions fit into the particular setup of stable homotopy.

Any object $E$ of any closed monoidal category $(\mathscr S,\bigwedge,[\_,\_])$ determines an adjoint pair of functors $E\bigwedge\_\dashv[E,\_]$ and thus both a monad $[E,E\bigwedge\_]$ and a comonad $E\bigwedge[E,\_]$ on $\mathscr S$. This then gives an adjoint pair between $E\bigwedge[E,\_]$-coalgebras and $[E,E\bigwedge\_]$-algebras and one may repeat ad infinitum to (hopefully) finally get some "$E$-local/stable/complete" category. In really good cases it is well related (although rarely equivalent) to $[E,E]$-modules but I do not know any good description in general.

Another version (which I learned from Claudio Hermida years ago) would involve adjunctions between (co)algebras and (co)Kleisli categories, rather than coalgebras and algebras. Since the Kleisli construction provides equivalents to the categories of (co)free (co)algebras, this version might be viewed as sort of approximations to the "no-relations" or "field-like" case where "all (co)algebras are (co)free". And if this cannot be achieved to the full, there might be calculable obstructions to doing it.

NB For this to actually work one in fact needs some amount of (co)equalizers; I wonder if (co)fibres would work in the triangulated setting...

If this construction would relate well to the "real thing" this would provide for the category-theorist part of me a nice motivation. Does it?

PS. There is yet another way to produce algebras/coalgebras via the $\textit{contravariant}$ adjunction $[\_,E]\dashv[\_,E]$ (this works in more general setting of a (not necessarily monoidal) closed category) and I might repeat the same question in this context too.