From a662e0a5212c92b15c6e84cf2907c5463e60da20 Mon Sep 17 00:00:00 2001
From: Philip Kaludercic <philip.kaludercic@fau.de>
Date: Sun, 7 Apr 2024 21:04:53 +0200
Subject: [PATCH] Clarify the definition of behavioural equivalence

---
 coalgebra.tex | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

diff --git a/coalgebra.tex b/coalgebra.tex
index 0df45f0..0f57783 100644
--- a/coalgebra.tex
+++ b/coalgebra.tex
@@ -50,10 +50,11 @@
 
 \begin{definition}\label{def:beheqiv}
   For a \de{endofunctor} \(F : \map{\Set}{\Set}\) and two \fca{}
-  \((C, c)\), \((D, d)\), are \emph{behaviourally equivalent} for
-  \(C \ni x \sim y \in D\) if
-  \[ \exists h, k \st (C, c) \xrightarrow{h} (E, e) \xleftarrow{k} (D,
-    d), \] for some \((E, e)\).
+  \((C, c)\), \((D, d)\) states \(x \in C\), \(y \in D\), are
+  \emph{behaviourally equivalent}, if for some
+  \((E, e)\),
+  \[ x \sim y \iff \exists h, k \st (C, c) \xrightarrow{h} (E, e)
+    \xleftarrow{k} (D, d). \]
 \end{definition}
 
 \begin{definition}\label{def:bisimulation}
-- 
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