Comparing tempered and equivariant elliptic cohomology

Lurie and Gepner–Meier each define equivariant cohomology theories, namely \emph{tempered cohomology} and \emph{equivariant elliptic cohomology}, respectively, using derived algebraic geometry. We construct a natural equivalence between these theories where they overlap. Moreover, we emphasise the naturality and coherence of both these equivariant theories as well as our comparison. To demonstrate the use of this comparison, we show that the $G$-fixed points of equivariant topological modular forms is dualisable as a $\mathrm{T}\mathrm{M}\mathrm{F}$-module for all compact Lie groups $G$ that decompose as a product of a torus and a finite group by formally reducing to an argument of Gepner–Meier.