Let $sSet = Set^{\Delta^{op}}$, $cSet = Set^{\square^{op}}$, $ccSet = Set^{\square\_c^{op}}$ be the categories of simplicial sets, cubical sets, and cubical sets with connections, respectively. Then there is a left adjoint functor $cSet \to sSet$ such that a representable cubical set $\square\_n$ is mapped to $\Delta[1]^n$. But I'm looking for a left adjoint functor going in the other direction.

We could try to define a functor $\Delta \to cSet$ in such a way that $\Delta[n]$ is mapped to a quotient of $\square\_n$. For example, $\Delta[2]$ is mapped to the square in which one of the faces is degenerated. This does not work since we cannot define a map of cubical sets corresponding to one of the degeneracy maps $\Delta[2] \to \Delta[1]$. But it seems that we can do this if we assume that our cubical sets have connections.

So, the question is: can we construct a (nontrivial) functor $\Delta \to ccSet$? Is there such a construction in the literature?