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@@ -2011,6 +2011,8 @@ So, now we see that for every natural relator $\relar$, a relation $(X \stackrel
\todo{Try to define simulation in a given double category. Perhaps you need to order enrichment over the endofunctor on $\BC_0$. Then inspired by~\autoref{prop:HeJ-HuJ} you may be able to prove a general theorem for an arbitrary lifting!} \todo{Try to define simulation in a given double category. Perhaps you need to order enrichment over the endofunctor on $\BC_0$. Then inspired by~\autoref{prop:HeJ-HuJ} you may be able to prove a general theorem for an arbitrary lifting!}
\section{Symmetric Simulation is a Bisimulation} \section{Symmetric Simulation is a Bisimulation}
\todo{Obviously, this chapter should be changed. All the definitions should be moved to somewhere else. You should start the chapter by giving your counter examples, and then presenting your proofs.} \todo{Obviously, this chapter should be changed. All the definitions should be moved to somewhere else. You should start the chapter by giving your counter examples, and then presenting your proofs.}
\todo{I think that these "double coalgebra"s for a category. Perhaps a double category! I wonder what is the final object there.
Additionally, you can think of defining "double algebras" and see the relevance with congruence relations!}
\begin{definition}[Graph] \begin{definition}[Graph]
In a category $\BC$ a graph is a tuple $(R,X)$ of the following form: In a category $\BC$ a graph is a tuple $(R,X)$ of the following form:
\begin{equation*} \begin{equation*}