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" category $L_E$. In really good cases it is well related to $[E,E]$-modules but I do not know any good description in general.
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. Is 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.