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