  We present explicit mathematical structures that allow for the reconstruction
of the field content of a full local conformal field theory from its boundary
fields. Our framework is the one of modular tensor categories, without
requiring semisimplicity, and thus covers in particular finite rigid
logarithmic conformal field theories. We assume that the boundary data are
described by a pivotal module category over the modular tensor category, which
ensures that the algebras of boundary fields are Frobenius algebras. Bulk
fields and, more generally, defect fields inserted on defect lines, are given
by internal natural transformations between the functors that label the types
of defect lines. We use the theory of internal natural transformations to
identify candidates for operator products of defect fields (of which there are
two types, either along a single defect line, or accompanied by the fusion of
two defect lines), and for bulk-boundary OPEs. We show that the so obtained
OPEs pass various consistency conditions, including in particular all
genus-zero constraints in Lewellen's list.
