acts on the indecomposable objects $E(X)$ by $$\Sigma( E( x_0,x_1 ) ) = E( x_0^-,x_1^- ).$$

5.  We have $$\label{eqn:Ext-spaces}
      \Ext_{ {\mathscr{C}}({\mathscr{Z}}) }^1( E(X),E(Y) ) \cong
        \begin{cases}
            k \text{ if $X$ and $Y$ cross,}\\
            0 \text{ otherwise. }
        \end{cases}$$

6.  Since $\Hom_{{\mathscr{C}}({\mathscr{Z}})}(E(X),E(Y)) \cong \Ext_{{\mathscr{C}}({\mathscr{Z}})}^1(E(X),\Sigma^{-1}E(Y))$, it follows from (iv) and (v) that $\Hom_{ {\mathscr{C}}({\mathscr{Z}}) }( E(X),E(Y) )$ is isomorphic to $$\begin{cases}
            k \text{ if we can write $X = \{ x_0,x_1 \}$ and $Y = \{ y_0,y_1 \}$ with $x_0 \leqslant y_0 \leqslant x_1^{ -- } < x_1 \leqslant y_1 \leqslant x_0^{--}$, }\\
            0 \text{ otherwise. }
        \end{cases}$$

7.  In part (vi), if $X = \{ x_0,x_1 \}$ and $Y = \{ y_0,y_1 \}$ with $x_0 \leqslant y_0 \leqslant x_1^{ -- } < x_1 \leqslant y_1 \leqslant x_0^{--}$, then a morphism $E(X) \to E(Y)$ factors through $E(S)$ if and only if we can write $S = \{ s_0,s_1 \}$ with $x_0 \leqslant s_0 \leqslant y_0$ and $x_1 \leqslant s_1 \leqslant y_1$.

In part (v) observe that non-vanishing of $\Ext^1$ is symmetric in the two arguments, as indeed it must be by the $2$-Calabi–Yau property from (ii). Figure \[fig:morphisms\] provides an illustration of morphisms between indecomposable objects.

(0,0) circle(0.5cm); at (60:0.58) [$x_0$]{}; 