  We study the values of the higher dimensional cardinal characteristics for
sets of functions $f:\omega^\omega \to \omega^\omega$ introduced by the second
author. We prove that while the bounding numbers for these cardinals can be
strictly less than the continuum, the dominating numbers cannot. We compute the
bounding numbers for the higher dimensional relations in many well known models
of $\neg\mathsf{CH}$ such as the Cohen, random and Sacks models and, as a
byproduct show that, with possibly one exception, for the bounding numbers
there are no $\mathsf{ZFC}$ relations between them beyond those in the higher
dimensional Cicho\'{n} diagram. In the case of the dominating numbers we show
that in fact they collapse in the sense that modding out by the ideal does not
change their values. Moreover, they are closely related to the dominating
numbers $\mathfrak{d}^\lambda_\kappa$.
