There are only id morphisms in \$$\mathcal{J} \$$ which is isomorphic to \$$\textbf{2} \$$.

$V_{\mathcal{J}} = \lbrace v, w \rbrace \\\\ A_{\mathcal{J}} = \lbrace 1_v, 1_w \rbrace$
$X \times Y = \lim_{\mathcal{J}} D := \lbrace (D(v), D(w)), (D(1_v), D(1_w)) \rbrace$
$p_v : X \times Y \mapsto D(v) \\\\ p_w : X \times Y \mapsto D(w);$