On Frobenius (completed) orbit categories

Let $\mathcal{E}$ be a Frobenius category, $\mathcal{P}$ its subcategory of projective objects and $F:\mathcal{E}\to \mathcal{E}$ an exact automorphism. We prove that there is a fully faithful functor from the orbit category $\mathcal{E}\mathrm{/}F$ into $\mathrm{gpr}(\mathcal{P}\mathrm{/}F)$, the category of finitely-generated Gorenstein-projective modules over $\mathcal{P}\mathrm{/}F$. We give sufficient conditions to ensure that the essential image of $\mathcal{E}\mathrm{/}F$ is an extension-closed subcategory of $\mathrm{gpr}(\mathcal{P}\mathrm{/}F)$. If $\mathcal{E}$ is in addition Krull-Schmidt, we give sufficient conditions to ensure that the completed orbit category $\mathcal{E}\widehat{\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{/}}F$ is a Krull-Schmidt Frobenius category. Finally, we apply our results on completed orbit categories to the context of Nakajima categories associated to Dynkin quivers and sketch applications to cluster algebras.