  We categorify the commutation of Nakajima's Heisenberg operators $P_{\pm 1}$
and their infinitely many counterparts in the quantum toroidal algebra
$U_{q_1,q_2}(\ddot{gl_1})$ acting on the Grothendieck groups of Hilbert
schemes. By combining our result with arxiv:1804.03645 , one obtains a
geometric categorical $U_{q_1,q_2}(\ddot{gl_1})$ action on the derived category
of Hilbert schemes. Our main technical tool is a detailed geometric study of
certain nested Hilbert schemes of triples and quadruples, through the lens of
the minimal model program, by showing that these nested Hilbert schemes are
either canonical or semi-divisorial log terminal singularities.
