  The notion of a d-critical locus is an ingredient in the definition of
motivic Donaldson-Thomas invariants by [BJM19]. There is a canonical d-critical
locus structure on the Hilbert scheme of dimension zero subschemes on local
toric Calabi-Yau 3-folds. This is obtained by truncating the $-1$-shifted
symplectic structure on the derived moduli stack [BBBBJ15]. In this paper we
show the canonical d-critical locus structure has critical charts consistent
with the description of Hilbert scheme as a degeneracy locus [BBS13]. In
particular, the canonical d-critical locus structure is isomorphic to the one
constructed in [KS12] for local $\mathbb{P}^2$ and local $\mathbb{F}_n$.
