makeUnderlyingFreeMackeyFunctor -- constructs the free Mackey functor on an underlying generator

The free $C_p$-Mackey functor on an underlying generator represents the functor $\text{Mack}_{C_p} \to \text{Ab}$ which sends a Mackey functor $M$ to its underlying level $M(C_p/e)$. This means there is a natural isomorphism $$\text{Hom}_{\text{Mack}_{C_p}}\left(\underline{B},M \right) \cong M(C_p/e).$$

In components, the underlying module is the free module on the $C_p$-set $C_p/e=\{1,\gamma,\gamma^2,\ldots,\gamma^{p-1}\}$, with conjugation induced by the $C_p$-action of left multiplication. The fixed module is the module $\ZZ$. Restriction is the map $$\text{res}_e^{C_p} \colon \ZZ \to \ZZ\{1,\gamma,\ldots,\gamma^{p-1}\}$$ by sending $1\mapsto 1+\gamma+\cdots+\gamma^{p-1}$. The transfer is the map $$\text{tr}_e^{C_p} \colon \ZZ\{1,\gamma,\ldots,\gamma^{p-1}\} \to \ZZ$$ by sending $\gamma^i\mapsto 1$ for all $i$.