  The correspondence between monoidal categories and graphical languages of
diagrams has been studied extensively, leading to applications in quantum
computing and communication, systems theory, circuit design and more. From the
categorical perspective, diagrams can be specified using (name-free)
combinators which enjoy elegant equational properties. However, conventional
notations for diagrammatic structures, such as hardware description languages
(VHDL, Verilog) or graph languages (Dot), use a different style, which is flat,
relational, and reliant on extensive use of names (labels). Such languages are
not known to enjoy nice syntactic equational properties. However, since they
make it relatively easy to specify (and modify) arbitrary diagrammatic
structures they are more popular than the combinator style. In this paper we
show how the two approaches to diagram syntax can be reconciled and unified in
a way that does not change the semantics and the existing equational theory.
Additionally, we give sound and complete equational theories for the combined
syntax.
