Proposition [lcc-002P]

The localization (resp. colocalization) \mathsf {Conc}^\mathrm {op} \hookrightarrow \mathsf {Minmax} (\mathsf {Conv} \hookrightarrow \mathsf {Minmax}) is monoidal, in the sense that the class local equivalences is stable under tensor products. The thus induced monoidal structure on \mathsf {Conc}^\mathrm {op} is given by (X,f) \otimes (Y,g) = (X \times Y, (x,y) \mapsto f(x) + g(y)) (and the same for \mathsf {Conv}). In particular the (co)localization functor is strong monoidal.