A note on \(\Gamma_G\)-spaces (Q1191031)

For a finite group \(G\), let \(\Gamma_G\) denote the category of based finite \(G\)-sets and based maps, where \(\Gamma_ G\) is endowed with the standard action \((q,f)\mapsto qfg^{-1}\) for \(g\in G\), \(f\in\text{mor }\Gamma_ G\). A \(\Gamma_ G\)-space is a \(G\)-equivariant functor from \(\Gamma_ G\) to the category of the based \(G\)-sets and based maps equipped with the standard \(G\)-action. A \(\Gamma\)-\(G\)-space is a functor from \(\Gamma=\Gamma e\) (\(e\) the trivial group) to the category of based \(G\)-spaces and based \(G\)-maps. Let \(\Gamma_ G\)-Space (resp. \(\Gamma\)- \(G\)-Space) denote the category of \(\Gamma_ G\)-spaces (resp. \(\Gamma\)- \(G\)-spaces) with \(G\)-equivariant natural transformations (resp. natural transformations) as morphisms. In this paper the author proves that there is an adjoint equivalence between \(\Gamma_G\)-Space and \(\Gamma\)-\(G\)-Space. Hence these two notions are canonically equivalent.