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In these notes we describe models of globular weak $(\infty,m)$-categories($m\in\mathbb{N}$) in the Grothendieck style, i.e for each $m\in\mathbb{N}$ wedefine a globular coherator $\Theta^{\infty}_{\mathbb{M}^m}$ whose set-modelsare globular weak $(\infty,m)$-categories. Then we describe the combinatoricsof the small category $\Theta_0$ whose objects are cubical pasting diagrams andwhose morphisms are morphisms of cubical sets. This provides an accuratedescription of the monad, on the category of cubical sets (without degeneraciesand connections), of cubical strict $\infty$-categories with connections. Weprove that it is a cartesian monad, solving a conjecture in\cite{camark-cub-1}. This puts us in a position to describe the cubicalcoherator $\Theta^{\infty}_W$ whose set-models are cubical weak$\infty$-categories with connections and the cubical coherator$\Theta^{\infty}_{W^{0}}$ whose set-models are cubical weak $\infty$-groupoidswith connections.

Combinatorial approach to the category $\Theta_0$ of cubical pasting diagrams