What I call "Spanier-Whitehead stabilization" is a construction which extends a category $\bf C$ to a bigger one $\mathcal{SW}_\Omega({\bf C})$ where a given endofunctor $\Omega$ is invertible. The category $\mathcal{SW}_\Omega({\bf C})$ is constructed with

- Objects the pairs $(A,n)\in Ob(\mathbf C)\times\mathbb Z$;
- The set of morphisms $(A,n)\to (B,m)$ corresponds to the colimit set $$ \varinjlim_{k\in\mathbb N} \hom_{\bf C}(\Omega^{n+k}A, \Omega^{m+k}B) $$

Now, whichever construction you choose for the functor $\mathcal{SW}\colon \mathbf{Cat}_E\to \mathbf{Cat}_A$ (there are many, using either monads or a direct construction reminding topology), what you have shown is that the full subcategory $\mathbf{Cat}_A$ of "categories with automorphism" is reflective in $\mathbf{Cat}_E$, "categories with *endo*morphism": the unit of this adjunction gives that you can draw a full arrow instead of a dotted one in

where $T\colon \mathcal B\cong\mathcal B$. I would like to generalize a bit this construction:

- Consider the category $\mathbf{Cat}_F$ of those categories ${\bf C}$ with an endofunctor $S$ such that there is $F\dashv S\dashv F$ ("Frobenius" adjunctions)
- Consider the category $\mathbf{Cat}_F$ of those categories $\bf C$ with an endofunctor $S$ such that there are $F\dashv S\dashv G$.

Are these two categories reflective in $\mathbf{Cat}_E$ too? If yes, how do you characterize the "Frobenification" and "haveanadjointonbothsides-ification" functors?