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We generalize the PL intersection product for chains on PL manifolds and forintersection chains on PL stratified pseudomanifolds to products of locallyfinite chains on non-compact spaces that are natural with respect torestriction to open sets. This is necessary to sheafify the intersectionproduct, an essential step in proving duality between the Goresky-MacPhersonintersection homology product and the intersection cohomology cup productpairing recently defined by the author and McClure.  We also provide a correction to the Goresky-MacPherson proof of a version ofPoincar\'e duality on pseudomanifolds that is used in the construction of theintersection product.

The chain-level intersection pairing for PL pseudomanifolds revisited