Direct limits and inverse limits of Mackey functors

Let G   be a finite group, SS the subgroup category of G, and M a Mackey functor on G  . For full subcategories GG and HH of SS, we have the direct limit M⁎(G)M⁎(G) and the inverse limit M⁎(H)M⁎(H) of M  . In this paper we study relation between the canonical homomorphisms ind:M⁎(G)→M(G)ind:M⁎(G)→M(G) and res:M(G)→M⁎(H)res:M(G)→M⁎(H).