  We revisit the construction by Sanders [S1] of the Hartle-Hawking-Israel
state for a free quantum Klein-Gordon field on a spacetime with a static,
bifurcate Killing horizon and a wedge reflection. Using the notion of the
Calder{\'o}n projector for elliptic boundary value problems and
pseudodifferential calculus on manifolds, we give a short proof of its Hadamard
property.
