  Composite basis functions for pyramidal elements on the spaces $H^1(\Omega)$,
$H(\mathrm{curl},\Omega)$, $H(\mathrm{div},\Omega)$ and $L^2(\Omega)$ are
presented. In particular, we construct the lowest-order composite pyramidal
elements and show that they respect the de Rham diagram, i.e. we have an exact
sequence and satisfy the commuting property. Moreover, the finite elements are
fully compatible with the standard finite elements for the lowest-order
Raviart-Thomas-N\'ed\'elec sequence on tetrahedral and hexahedral elements.
That is to say, the new elements have the same degrees of freedom on the shared
interface with the neighbouring hexahedral or tetrahedra elements, and the
basis functions are conforming in the sense that they maintain the required
level of continuity (full, tangential component, normal component, ...) across
the interface. Furthermore, we study the approximation properties of the spaces
as an initial partition consisting of tetrahedra, hexahedra and pyramid
elements is successively subdivided and show that the spaces result in the same
(optimal) order of approximation in terms of the mesh size $h$ as one would
obtain using purely hexahedral or purely tetrahedral partitions.
