Short injective proofs of the Erdős-Ko-Rado and Hilton-Milner Theorem: A canonical partition of shifted intersecting set systems

We give a canonical partition of shifted intersecting set systems, from which one can obtain unified and elementary proofs of the Erdős-Ko-Rado and Hilton-Milner Theorem, as well as a characterization of maximal shifted $k$-uniform intersecting set systems over $[n]$.