Coincidence of Schur multipliers of the Drury-Arveson space

In a purely multi-variable setting (i.e., the issues discussed in this note are not interesting in the single-variable operator theory setting), we show that the coincidence of two operator-valued Schur class multipliers of a certain kind on the Drury-Arveson space is characterized by the fact that the associated colligations (or a variant, obtained canonically) are 'unitarily coincident' in a sense to be made precise in this article.