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In this expository note, we compare the fusion product of conformal fieldtheory, as defined by Gaberdiel and used in the Nahm-Gaberdiel-Kausch (NGK)algorithm, with the $P(w)$-tensor product of vertex operator algebra modules,as defined by Huang, Lepowsky and Zhang (HLZ). We explain how the equality ofthe two "coproducts" derived by NGK is essentially dual to the$P(w)$-compatibility condition of HLZ and how the algorithm of NGK forcomputing fusion products may be adapted to the setting of HLZ. We provideexplicit calculations and instructive examples to illustrate both approaches.This document does not provide precise descriptions of all statements, it isintended more as a gentle starting point for the appreciation of the depth ofthe theory on both sides.

NGK and HLZ: fusion for physicists and mathematicians