  Around the year 1988, Joyal and Street established a graphical calculus for
monoidal categories, which provides a firm foundation for many explorations of
graphical notations in mathematics and physics. For a deeper understanding of
their work, we consider a similar graphical calculus for semi-groupal
categories. We introduce two frameworks to formalize this graphical calculus, a
topological one based on the notion of a processive plane graph and a
combinatorial one based on the notion of a planarly ordered processive graph,
which serves as a combinatorial counterpart of a deformation class of
processive plane graphs. We demonstrate the equivalence of Joyal and Street's
graphical calculus and the theory of upward planar drawings. We introduce the
category of semi-tensor schemes, and give a construction of a free monoidal
category on a semi-tensor scheme. We deduce the unit convention as a kind of
quotient construction, and show an idea to generalize the unit convention.
Finally, we clarify the relation of the unit convention and Joyal and Street's
construction of a free monoidal category on a tensor scheme.
