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