--- a/text/appendixes/comparing_defs.tex	Wed Jan 27 19:34:48 2010 +0000
+++ b/text/appendixes/comparing_defs.tex	Mon Feb 01 06:11:18 2010 +0000
@@ -18,7 +18,7 @@
 Choose a homeomorphism $B^1\cup_{pt}B^1 \to B^1$.
 Define composition in $c(\cX)$ to be the induced map $c(\cX)^1\times c(\cX)^1 \to c(\cX)^1$ (defined only when range and domain agree).
 By isotopy invariance in $\cX$, any other choice of homeomorphism gives the same composition rule.
-Also by isotopy invariance, composition is associative on the nose.
+Also by isotopy invariance, composition is strictly associative.
 
 Given $a\in c(\cX)^0$, define $\id_a \deq a\times B^1$.
 By extended isotopy invariance in $\cX$, this has the expected properties of an identity morphism.
@@ -126,7 +126,7 @@
 Given $x\in C^0$, define $\id_x = x\times B^1 \in C^1$.
 We will show that this 1-morphism is a weak identity.
 This would be easier if our 2-morphisms were shaped like rectangles rather than bigons.
-Define let $a: y\to x$ be a 1-morphism.
+Let $a: y\to x$ be a 1-morphism.
 Define maps $a \to a\bullet \id_x$ and $a\bullet \id_x \to a$
 as shown in Figure \ref{fzo2}.
 \begin{figure}[t]
@@ -137,7 +137,7 @@
 \label{fzo2}
 \end{figure}
 In that figure, the red cross-hatched areas are the product of $x$ and a smaller bigon,
-while the remained is a half-pinched version of $a\times I$.
+while the remainder is a half-pinched version of $a\times I$.
 \nn{the red region is unnecessary; remove it?  or does it help?
 (because it's what you get if you bigonify the natural rectangular picture)}
 We must show that the two compositions of these two maps give the identity 2-morphisms