Let $\Delta_2$ the 2-tronked simplicial category (objects are the finite orders $0=$(0), $1=$(0,1), $2=$(0, 1, 2) and order-preserving functions). Then $Cat$ (category of small categories)is isomorphically to $Fun_c(\Delta_2^{op}, Set)$ (finite limits preserving functors and natural transformations).
More in general the (hyper)category of n-double small categories $n$-$Double$ is isomorphically to $Fun_c((\Delta_2^{op}\times\ldots\times\Delta_2^{op}), Set)$ (n fold products), and $n$-$Cat$ is a subcategory of $n$-$Double$ (elements of $Fun_c((\Delta_2^{op}\times\ldots\times\Delta_2^{op}), Set)$ that send some morphisms to identities). Of course you can generalize this for a infinite cardinal $n$.