Realising the c*-algebra of a higher-rank graph as an exel's crossed product

We use the boundary-path space of a finitely-aligned k-graph Lambda to construct a compactly-aligned product system X, and we show that the graph algebra C*(Lambda) is isomorphic to the Cuntz-Nica-Pimsner algebra NO(X). In this setting, we introduce the notion of a crossed product by a semigroup of partial endomorphisms and partially-defined transfer operators by defining it to be NO(X). We then compare this crossed product with other definitions in the literature.