Fix "a representation is a universal element"

trees/tt-0035.tree

@@ -11,28 +11,26 @@
 % kostecki2011introduction leinster2016basic
 
 \taxon{corollary}\refnotet{a representation is a universal element}{4.3.2}{leinster2016basic}{
-\p{Let #{\C} be a locally small category and #{\fF : \C^{op} \to \Set}. Then a \vocabk{representation}{tt-001O} of #{\fF} consists of an object #{X \in \C} together with an element #{u \in \fF(X)} such that:
-}
-
-\p{for each #{Y \in \C} and #{y \in \fF(Y)}, there is a unique map #{\bar{y}: Y \to X} such that #{(\fF \bar{y})(u)=y}.
+\p{Let #{\C} be a locally small category and #{\fF : \C^{op} \to \Set}. Then a \vocabk{representation}{tt-001O} of #{\fF} consists of a pair #{(X, u)} such that the diagram
 \tikz{
 \begin{tikzcd}
-	&&&& X \\
-	X &&&& O \\
+	X &&&&& {\C(X, O)} \\
 	\\
+	&&& O \\
 	\\
-	Y &&&& Y \\
-	&&&& O \\
-	\C &&&& \Set
-	\arrow[""{name=0, anchor=center, inner sep=0}, "u"', from=1-5, to=2-5]
-	\arrow[""{name=1, anchor=center, inner sep=0}, "{{\bar{y}}}", dashed, from=5-1, to=2-1]
-	\arrow[""{name=2, anchor=center, inner sep=0}, "y"', from=5-5, to=6-5]
-	\arrow["{{\fF : \C^{op} \to \Set, - \mapsto \C(-, O)}}"', Rightarrow, from=7-1, to=7-5]
-	\arrow[""{name=3, anchor=center, inner sep=0}, "{{\bar{y} \cp -}}"', curve={height=-30pt}, shorten <=10pt, shorten >=10pt, maps to, from=0, to=2]
-	\arrow[shorten <=24pt, shorten >=24pt, Rightarrow, maps to, from=2-1, to=0]
-	\arrow[shorten <=24pt, shorten >=24pt, Rightarrow, maps to, from=5-1, to=2]
-	\arrow[shorten <=32pt, shorten >=32pt, Rightarrow, maps to, from=1, to=3]
+	Y &&&&& {\C(Y, O)} \\
+	{\C^{op}} &&&&& \Set
+	\arrow[""{name=0, anchor=center, inner sep=0}, "u"{description}, curve={height=-24pt}, from=1-1, to=3-4]
+	\arrow[""{name=1, anchor=center, inner sep=0}, "{\bar{y} \cp -}"{pos=0.4}, from=1-6, to=5-6]
+	\arrow[""{name=2, anchor=center, inner sep=0}, "{{{\bar{y}}}}"{pos=0.6}, dashed, from=5-1, to=1-1]
+	\arrow[""{name=3, anchor=center, inner sep=0}, "y"{description}, curve={height=24pt}, from=5-1, to=3-4]
+	\arrow["{\fF : \C(-,O)}"', Rightarrow, from=6-1, to=6-6]
+	\arrow["\in"{description}, draw=none, from=0, to=1-6]
+	\arrow[shift left=5, curve={height=-12pt}, shorten <=17pt, shorten >=17pt, Rightarrow, from=2, to=1]
+	\arrow["\in"{marking, allow upside down}, draw=none, from=3, to=5-6]
 \end{tikzcd}
 }
+
+commutes.
 }
 }