diff --git a/trees/tt-000J.tree b/trees/tt-000J.tree
index 77728d7..5d16f73 100644
--- a/trees/tt-000J.tree
+++ b/trees/tt-000J.tree
@@ -5,11 +5,11 @@
 % definition theorem lemma construction observation
 % convention corollary axiom example exercise proof
 % discussion remark notation
-\taxon{notation}
+\taxon{convention}
 
-\title{Uniqueness}
+\title{uniqueness: dashed arrow}
 
-\p{Uniqueness of an arrow is denoted #{\exists ! f} or simply #{!f}, and visualized as a dashed arrow in diagrams, and #{!} is often omitted.}
+\p{Uniqueness of an arrow is denoted #{\exists ! f} or simply #{!f}, and visualized as a \vocab{dashed arrow} in diagrams, and #{!} is often omitted.}
 
 \tikz{
 \begin{tikzcd}
diff --git a/trees/tt-000O.tree b/trees/tt-000O.tree
index 50c97c2..241f044 100644
--- a/trees/tt-000O.tree
+++ b/trees/tt-000O.tree
@@ -9,7 +9,7 @@
 \refdef{fork}{leinster2016basic}{
 \p{
 
-  Two objects with two arrows between them
+  The diagram
 
   \tikz{
 \begin{tikzcd}
@@ -20,6 +20,9 @@
   } is called a \newvocab{fork}.
 
 }
+
+\p{For simplicity, we refer to a fork by "a fork #{(f, g)}".}
+
 }
 
 
diff --git a/trees/tt-000P.tree b/trees/tt-000P.tree
index 7d30972..e76b895 100644
--- a/trees/tt-000P.tree
+++ b/trees/tt-000P.tree
@@ -13,5 +13,8 @@
 
 \transclude{tt-000O}
 
-\transclude{tt-000Q}
+\transclude{tt-000R}
+
+\transclude{tt-000S}
 
+\transclude{tt-000Q}
diff --git a/trees/tt-000Q.tree b/trees/tt-000Q.tree
index 51cf362..2580a65 100644
--- a/trees/tt-000Q.tree
+++ b/trees/tt-000Q.tree
@@ -9,7 +9,7 @@
 
 \refdef{equalizer}{kostecki2011introduction}{
 \p{
-An \newvocab{equalizer} of two given arrows #{f, g: A \rightarrow B} is an object #{E} together with a morphism #{e: E \rightarrow A} such that #{f \circ e=g \circ e}, and for any object #{D} and morphism #{h: D \rightarrow A} there exists a unique morphism #{k: D \rightarrow E} such that the diagram
+An \newvocab{equalizer} of a fork #{(f, g)} in the same \vocab{fork} is an object #{E} together with a arrow #{e: E \rightarrow A} such that #{f \circ e=g \circ e}, and for any object #{D} and arrow #{h: D \rightarrow A}, the diagram
 
 \tikz{
 \begin{tikzcd}
diff --git a/trees/tt-000R.tree b/trees/tt-000R.tree
new file mode 100644
index 0000000..b7072f5
--- /dev/null
+++ b/trees/tt-000R.tree
@@ -0,0 +1,29 @@
+\import{tt-macros}
+% clifford hopf spin tt math draft
+\tag{tt}
+
+% definition theorem lemma construction observation
+% convention corollary axiom example exercise proof
+% discussion remark notation
+% \taxon{}
+
+\refdef{cone}{leinster2016basic}{
+\p{
+A \newvocab{cone} over a \vocab{fork} #{(f, g)} is an object #{E} and arrows over the fork which make the diagram
+
+\tikz{
+  \begin{tikzcd}
+    & E \\
+    X && Y
+    \arrow["e"', from=1-2, to=2-1]
+    \arrow["{e;f = e;g}", curve={height=-6pt}, dotted, from=1-2, to=2-3]
+    \arrow["f", shift left, from=2-1, to=2-3]
+    \arrow["g"', shift right, from=2-1, to=2-3]
+  \end{tikzcd}
+}
+
+ commute.
+}}
+
+\p{For simplicity, we refer to a cone by "a cone #{e}".}
+
diff --git a/trees/tt-000S.tree b/trees/tt-000S.tree
new file mode 100644
index 0000000..87ec589
--- /dev/null
+++ b/trees/tt-000S.tree
@@ -0,0 +1,12 @@
+\import{tt-macros}
+% clifford hopf spin tt math draft
+\tag{tt}
+
+% definition theorem lemma construction observation
+% convention corollary axiom example exercise proof
+% discussion remark notation
+\taxon{convention}
+
+\title{dotted arrow}
+
+\p{We use \newvocab{dotted arrow}s to represent the composition arrow in a \vocab{cone}. This convention is not from the literature and is subject to change.}