  In this paper we study intrinsic regular submanifolds of $\mathbb{H}^n$, of
low co-dimension in relation with the regularity of their intrinsic
parametrization. We extend some results proved for one co-dimensional
$\mathbb{H}$-regular surfaces, characterizing uniformly intrinsic
differentiable functions $\phi$ acting between two complementary subgroups of
the Heisenberg group $\mathbb{H}^n$, with target space horizontal of dimension
$k$, with $1 \leq k \leq n$, in terms of the Euclidean regularity of its
components with respect to a family of non linear vector fields
$\nabla^{\phi_j}$. Moreover, we show how the area of the intrinsic graph of
$\phi$ can be computed through the component of the matrix identifying the
intrinsic differential of $\phi$.
