boxProduct -- box product of Cp-Mackey functors

Given two Cp-Mackey functors $M$ and $N$, we can form their box product, which creates another Mackey functor. This new Mackey functor is defined by the following data:

• underlying module: defined as $M(C_p/e) \otimes N(C_p/e)$
• fixed module: defined as $(M(C_p/C_p)\otimes N(C_p/C_p)) \oplus (M(C_p/e) \otimes N(C_p/e))/ \sim$ where the equivalence relation is generated by $$( \text{tr}(x)\otimes b,0 ) \sim ( 0, x\otimes\text{res}(b))$$$$(a\otimes\text{tr}(y),0)\sim(0,\text{res}(a)\otimes y)$$ and $$(0, x\otimes y)\sim(0, c(x)\otimes c(y))$$
• restriction: defined by the formula $$[a\otimes b, x\otimes y] \mapsto \text{res}(a) \otimes \text{res}(b) + \sum_{i=0}^{p-1} c^i(x) \otimes c^i(y).$$
• transfer: defined by the formula$$x\otimes y \mapsto [0,x \otimes y]$$
• conjugation: defined by $c\otimes c$.

The box product makes $\text{Mack}_{C_p}$ into a symmetric monoidal category, which is closed symmetric monoidal due to the internal hom.

i1 : M= makeUnderlyingFreeMackeyFunctor(2);

i2 : M**M

o2 =                                         Res : | 1 1 0 0 1 |
                                                   | 1 0 1 1 0 |
                                                   | 1 0 1 1 0 |
                                                   | 1 1 0 0 1 |
     cokernel | 1  1  1  1  0  0  0  0  |  ----------------------->   4  -┐ Conj : | 0 0 0 1 |
              | -1 0  -1 0  1  0  0  -1 | <-----------------------  ZZ   <┘        | 0 0 1 0 |
              | -1 0  0  -1 0  1  -1 0  |    Tr : | 0 1  1  0 |                    | 0 1 0 0 |
              | 0  -1 -1 0  0  -1 1  0  |         | 1 -1 -1 1 |                    | 1 0 0 0 |
              | 0  -1 0  -1 -1 0  0  1  |         | 0 0  0  0 |
                                                  | 0 0  0  0 |
                                                  | 0 0  0  0 |

o2 : CpMackeyFunctor