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. In order to construct the categorification, we construct two triple moduli spaces and study their singularity structure from the viewpoint of the minimal model program. We prove that one moduli space is a canonical singularity and the other one is semi-divisorial log terminal. It leads to the natural transformations in derived categories.