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In this article, we construct a 2-shaded rigid ${\rm C}^*$ multitensorcategory with canonical unitary dual functor directly from a standard$\lambda$-lattice. We use the notions of traceless Markov towers and latticesto define the notion of module and bimodule over standard $\lambda$-lattice(s),and we explicitly construct the associated module category and bimodulecategory over the corresponding 2-shaded rigid ${\rm C}^*$ multitensorcategory.  As an example, we compute the modules and bimodules for Temperley-Lieb-Jonesstandard $\lambda$-lattices in terms of traceless Markov towers and lattices.Translating into the unitary 2-category of bigraded Hilbert spaces, we recoverDeCommer-Yamshita's classification of $\mathcal{TLJ}$ modules in terms of edgeweighted graphs, and a classification of $\mathcal{TLJ}$ bimodules in terms ofbiunitary connections on square-partite weighted graphs.  As an application, we show that every (infinite depth) subfactor planaralgebra embeds into the bipartite graph planar algebra of its principal graph.

Standard $\lambda$-lattices, rigid ${\rm C}^*$ tensor categories, and (bi)modules