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We use intuitive results from algebraic topology and intersection theory toclarify the pullback action on cohomology by compositions of rational maps. Weuse these techniques to prove a simple sufficient criterion for functorialityof a composition of two rational maps on all degrees of cohomology and we thenreprove the criteria of Diller-Favre, Bedford-Kim, and Dinh-Sibony. We concludewith a cautionary example.

The action on cohomology by compositions of rational maps