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@@ -1021,7 +1021,7 @@ The given definition is highly abstract. There is a relation lifting that abstra
\end{cor} \end{cor}
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\subsection{Coalgebraic Bisimulation in Set} \subsection{Coalgebraic Bisimulation in Set}
We have two more notions for coalgebraic bisimulation in $\Set$, that is to define them in $\spa_a$ and $\rel_a$, respectively called \emph{span-based bisimulation} and \emph{relator-based bisimulation}. We have two more notions for coalgebraic bisimulation in $\Set$, that is to define them in $\spa_a$ and $\rel_a$, respectively called \emph{span-based bisimulation} and \emph{Hughes-Jacobs bisimulation}.
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\begin{definition}[Hughes-Jacobs Bisimulation] \begin{definition}[Hughes-Jacobs Bisimulation]
A relation $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ in $\rel_a$ is a \emph{Hughes-Jacobs bisimulation} over $F$-coalgebras $(X,\alpha)$ and $(Y,\beta)$, whenever there exists a morhpism in $\rel_a$ of the type $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)\to(FX \stackrel{(Fp_1)^\dagger}{\leftarrow} (FR)^\dagger \stackrel{(Fp_2)^\dagger}{\to}FY)$. A relation $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)$ in $\rel_a$ is a \emph{Hughes-Jacobs bisimulation} over $F$-coalgebras $(X,\alpha)$ and $(Y,\beta)$, whenever there exists a morhpism in $\rel_a$ of the type $(X \stackrel{p_1}{\leftarrow} R \stackrel{p_2}{\to}Y)\to(FX \stackrel{(Fp_1)^\dagger}{\leftarrow} (FR)^\dagger \stackrel{(Fp_2)^\dagger}{\to}FY)$.