the vertical segments or boxes. Finally, compose the resulting morphisms proceeding from the bottom to the top. For example, given $A,B,X,Y,Z \in {\mathrm{Ob}}({\mathcal{S}})$ and morphisms $f{\colon}Y \to Z$, $g{\colon}B \otimes Z \to {\mathbb{1}}$, $h {\colon}X \to A \otimes B$, the diagram $$\psfrag{h}[Bc][Bc]{\scalebox{.9}{$g$}}
\psfrag{k}[Bc][Bc]{\scalebox{.9}{$f$}}
\psfrag{b}[Bc][Bc]{\scalebox{.9}{$h$}}
\psfrag{X}[Bl][Bl]{\scalebox{.8}{$X$}}
\psfrag{Y}[Bl][Bl]{\scalebox{.8}{$Z$}}
\psfrag{A}[Bl][Bl]{\scalebox{.8}{$A$}}
\psfrag{B}[Bl][Bl]{\scalebox{.8}{$B$}}
\psfrag{Z}[Bl][Bl]{\scalebox{.8}{$Y$}}
{\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{Penrose-ex.eps}}}}$$ represents the morphism $$({\mathrm{id}}_A \otimes g \otimes {\mathrm{id}}_X)(h \otimes f \otimes {\mathrm{id}}_X)=\bigl ( ({\mathrm{id}}_A \otimes g)(h \otimes f) \bigr ) \otimes {\mathrm{id}}_X$$ from $X \otimes Y \otimes X$ to $A \otimes X$.

The functoriality of the monoidal product implies that the morphism associated to a ${\mathcal{S}}$-colored diagram is independent of the way of cutting it into horizontal strips. It also implies that we can push boxes lying on the same horizontal level up or down without changing the morphism represented by the diagram. For example, for all morphisms $f{\colon}X \to Y$ and $g{\colon}U \to
V$ in ${\mathcal{S}}$, we have: $$\psfrag{X}[Bl][Bl]{\scalebox{.8}{$X$}}
\psfrag{f}[Bc][Bc]{\scalebox{.9}{$f$}}
\psfrag{Y}[Bl][Bl]{\scalebox{.8}{$Y$}}
\psfrag{U}[Bl][Bl]{\scalebox{.8}{$U$}}
\psfrag{g}[Bc][Bc]{\scalebox{.9}{$g$}}
\psfrag{V}[Bl][Bl]{\scalebox{.8}{$V$}}
{\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{ex-penro6b.eps}}}} \, = \, {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{ex-penro7.eps}}}} \, = \, {\raisebox{-.45\height}{\scalebox{.9}{\includegraphics{ex-penro6.eps}}}}$$ which graphically expresses the formulas $$f\otimes g =({\mathrm{id}}_Y\otimes g) (f \otimes {\mathrm{id}}_U)= (f\otimes {\mathrm{id}}_V)( {\mathrm{id}}_X\otimes g).$$ Here and in the sequel, the equality sign between the diagrams means the equality of the corresponding morphisms.

The symmetry 