--- a/text/ncat.tex	Fri May 06 15:23:26 2011 -0700
+++ b/text/ncat.tex	Fri May 06 16:04:22 2011 -0700
@@ -124,10 +124,13 @@
 \end{lem}
 
 We postpone the proof of this result until after we've actually given all the axioms.
-Note that defining this functor for some $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, 
-along with the data described in the other axioms at lower levels. 
+Note that defining this functor for fixed $k$ only requires the data described in Axiom \ref{axiom:morphisms} at level $k$, 
+along with the data described in the other axioms for smaller values of $k$. 
 
-%In fact, the functors for spheres are entirely determined by the functors for balls and the subsequent axioms. (In particular, $\cC(S^k)$ is the colimit of $\cC$ applied to decompositions of $S^k$ into balls.) However, it is easiest to think of it as additional data at this point.
+Of course, Lemma \ref{lem:spheres}, as stated, is satisfied by the trivial functor.
+What we really mean is that there is exists a functor which interacts with other data of $\cC$ as specified 
+in the other axioms below.
+
 
 \begin{axiom}[Boundaries]\label{nca-boundary}
 For each $k$-ball $X$, we have a map of sets $\bd: \cC_k(X)\to \cl{\cC}_{k-1}(\bd X)$.