${\text{\tt{C}}}$-morphisms from $X$ to $Y$ that make the diagram on the right commute.

[r]{}[.1]{}

All base categories ${\text{\tt{C}}}$ in this paper will be *algebraic categories*, i.e., categories ${{\text{\tt{Set}}}^{T}}$ of Eilenberg-Moore algebras of a finitary monad [^1] in ${{\text{\tt{Set}}}}$. Hence, all base categories are concrete with forgetful functor that is identity on morphisms.

In such categories behavioural equivalence [@Kurz00:thesis; @Wol00:cmcs; @Staton11] can be defined as follows. Let $(X,c)$ and $(Y,d)$ be $F$-coalgebras and let $x\in X$ and $y\in Y$. Then $x$ and $y$ are *behaviourally equivalent*, and we write $x\sim y$, if there exists an $F$-coalgebra $(Z,e)$ and ${{\sf Coalg}({F})}$-morphisms $f\colon (X,c)\to(Z,e)$, $g\colon(Y,d)\to(Z,e)$, with $f(x)=g(y)$. $$\xymatrix@C=40pt{
        (X,c) \ar[r]^f
        & (Z,e) 
        \save[]+<0pt,-13pt>*\txt{$\scriptstyle f(x)=g(y)$}\restore
        & (Y,d) \ar[l]_g 
    }$$ If there exists a final coalgebra in ${{\sf Coalg}({F})}$, and all functors considered in this paper will have this property, then two elements are behaviourally equivalent if and only if they have the same image in the final coalgebra. If we have a *zig-zag diagram* in ${{\sf Coalg}({F})}$ $$\label{zig-zag}
    \xymatrix@R=0.7em@C=9pt{
        (X,c) \ar[rd]^{f_1} && (Z_2,e_2) \ar[ld]_{f_2} \ar[rd]^{f_3} && 
        \cdots \ar[ld]_{f_4} \ar[rd]^{f_{2n-1}} && (Y,d) \ar[ld]_{f_{2n}} &
        \\
        & (Z_1,e_1) 
        && (Z_3,e_1) 
        && (Z_{2n-1},e_1) 
        &
    }$$ which relates $x$ with $y$ in the sense that there exist elements 