I have been reading Categories for the Working Mathematician (Second Edition), and have come across what I think might be a typo (Expression (8) of page 280). I think this should be an equation, but there is no equals sign. I was hoping somebody would tell me what the correction to this expression is (or the interpretation). I will paraphrase the relevant section, and point out the part that confuses me..
A 2-category can be considered to be a single set $X$ of 2-cells, with extra structure (objects and arrows are treated as degenerate 2-cells). The set $X$ has a horizontal structure $(\#_0,s_0,t_0)$ and a vertical structure $(\#_1,s_1,t_1)$ such that for each $\alpha$ in $\{0,1\}$ and $x,y,z$ in $X$ we have:
$s_\alpha , t_\alpha : X \rightarrow X$ give the source and target of a 2-cell.
$\#_\alpha$ is a binary operation on $X$.
$x \#_\alpha y$ is defined iff $s_\alpha x = t_\alpha y$ and then
$s_\alpha (x \#_\alpha y) = s_\alpha y$
$t_\alpha (x \#_\alpha y) = t_\alpha x,$
$x \#_\alpha s_\alpha x = x$
$t_\alpha x \#_\alpha x = x$
$(x \#_\alpha y) \#_\alpha z = x \#_\alpha (y \#_\alpha z)$ if either side is defined,
$s_\alpha s_\alpha x = s_\alpha x = t_\alpha s_\alpha x$
$t_\alpha t_\alpha x = t_\alpha x = s_\alpha t_\alpha x$
(1) Every identity for the $0$-structure is an identity for the $1$-structure.
(2) The two structures commute with each other.
Mac Lane says (2) means that
$s_0 s_1= s_1 s_0, s_0 t_1 = t_1 s_0, t_0 s_1 = s_1 t_0, t_0 t_1 = t_1 t_0$
and that for $\alpha,\beta = 0,1$ or $1,0$ and all $x,y,u,v$ we have these three "conditions"
$t_\alpha (x \# _\beta y)$ = $(t_\alpha x) \# _\beta (t_\alpha y)$
$s_\alpha (x \# _\beta y)$ = $(s_\alpha x) \# _\beta (s_\alpha y)$
$(x \# _\alpha y) \#_\beta (u \#_\alpha v) \#_\alpha (y \#_\beta v).$
It is the last line that confuses me (corresponding to Mac Lane's Expression (8)), since it lacks an equals sign, and does not look like a proper condition. Did Mac Lane mean to state the interchange law $(x \# _\alpha y) \#_\beta (u \#_\alpha v) = (x \# _\beta u) \#_\alpha (y \#_\beta v)$ here ? Or something else ?
Additionally I am wondering how 2-functors and 2-natural transformations are defined in this type of setup. Any recommendations about further literature regarding this single-set approach to $n$-categories would also be most welcome.