--- a/text/appendixes/comparing_defs.tex	Tue Dec 13 07:44:47 2011 -0800
+++ b/text/appendixes/comparing_defs.tex	Tue Dec 13 07:57:01 2011 -0800
@@ -637,7 +637,7 @@
 
 Given 2-morphisms $f$ and $g$, we define the horizontal composition $f *_h g$ to be any of the four
 equal 2-morphisms in Figure \ref{fzo5}.
-Figure \ref{fig:horizontal-compositions-equal}illustrates part of the proof that these four 2-morphisms are equal.
+Figure \ref{fig:horizontal-compositions-equal} illustrates part of the proof that these four 2-morphisms are equal.
 Similar arguments show that horizontal composition is associative.
 \begin{figure}[t]
 \begin{align*}
@@ -926,7 +926,7 @@
 Figure \ref{fig:ingredients-triangle-axiom}.
 (See also Figures \ref{fzo2} and \ref{fig:associator}.)
 We now show that $D$ satisfies the triangle axiom, which states that $u\bullet\id_b$ 
-is equal to the composition of $\alpha$ and $\id_a\bullet v$.
+is equal to the vertical composition of $\alpha$ and $\id_a\bullet v$.
 (Both are 2-morphisms from $(a\bullet \id_y)\bullet b$ to $a\bullet b$.)
 \begin{figure}[t]
 %\begin{align*}
@@ -1016,10 +1016,10 @@
 \label{fig:ingredients-triangle-axiom}
 \end{figure}
 
-The horizontal compositions $u\bullet\id_b$ and $\id_a\bullet v$ are shown in Figure \ref{fig:horizontal-composition}
+The horizontal compositions $u *_h \id_b$ and $\id_a *_h  v$ are shown in Figure \ref{fig:horizontal-composition}
 (see also Figure \ref{fzo5}).
-The vertical composition of $\alpha$ and $\id_a\bullet v$ is shown in Figure \ref{fig:vertical-composition}.
-Figure \ref{fig:adding-a-collar} shows that we can add a collar to $u\bullet\id_b$ so that the result differs from
+The vertical composition of $\alpha$ and $\id_a *_h  v$ is shown in Figure \ref{fig:vertical-composition}.
+Figure \ref{fig:adding-a-collar} shows that we can add a collar to $u *_h \id_b$ so that the result differs from
 Figure  \ref{fig:vertical-composition} by an isotopy rel boundary.
 Note that here we have used in an essential way the associativity of product morphisms (Axiom \ref{axiom:product}.3) 
 as well as compatibility of product morphisms with fiber-preserving maps (Axiom \ref{axiom:product}.1).