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If K is a discrete group and Z is a K-spectrum, then the homotopy fixed pointspectrum Z^{hK} is Map_*(EK_+, Z)^K, the fixed points of a familiar expression.Similarly, if G is a profinite group and X is a discrete G-spectrum, thenX^{hG} is often given by (H_{G,X})^G, where H_{G,X} is a certain explicitconstruction given by a homotopy limit in the category of discrete G-spectra.Thus, in each of two common equivariant settings, the homotopy fixed pointspectrum is equal to the fixed points of an explicit object in the ambientequivariant category. We enrich this pattern by proving in a precise sense thatthe discrete G-spectrum H_{G,X} is just "a profinite version" of Map_*(EK_+,Z): at each stage of its construction, H_{G,X} replicates in the setting ofdiscrete G-spectra the corresponding stage in the formation of Map_*(EK_+, Z)(up to a certain natural identification).

Homotopy fixed points for profinite groups emulate homotopy fixed points for discrete groups