--- a/text/appendixes/comparing_defs.tex	Thu Aug 11 22:14:11 2011 -0600
+++ b/text/appendixes/comparing_defs.tex	Fri Aug 12 10:00:59 2011 -0600
@@ -585,6 +585,7 @@
 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$
 ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$).
 For simplicity we will now assume there is only one object and suppress it from the notation.
+Henceforth $A$ will also denote its unique morphism space.
 
 A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\otimes A\to A$.
 We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic.
@@ -610,7 +611,7 @@
 We define a $\Homeo(J)$ action on $\cC(J)$ via $g_*(f, a) = (g\circ f, a)$.
 The $C_*(\Homeo(J))$ action is defined similarly.
 
-Let $J_1$ and $J_2$ be intervals.
+Let $J_1$ and $J_2$ be intervals, and let $J_1\cup J_2$ denote their union along a single boundary point.
 We must define a map $\cC(J_1)\ot\cC(J_2)\to\cC(J_1\cup J_2)$.
 Choose a homeomorphism $g:I\to J_1\cup J_2$.
 Let $(f_i, a_i)\in \cC(J_i)$.