On the page $9$ here I have a very basic question: in the definition $3.1$ $\alpha$ never appears in the condition
"such that if $(U_1f)a\in U_2 HB$ then $a\in U_2 HA$ for each $f:A\to B$ in ${\cal K}_1$".
Also, I cannot syntactically verify this condition.
$\alpha$ is a monic natural transformation of functors to Set, so the authors identity the components of $\alpha$ with the subset relation. That is, for all $A\in \mathcal{K}_1$, the authors suppress the injective function $\alpha_A\colon U_2HA\hookrightarrow U_1A$ and instead pretend $U_2HA\subseteq U_1A$. This can be done formally at the expense of replacing $U_1$ or $U_2$ with a naturally isomorphic copy.
So the condition "if $(U_1f)a\in U_2HB$, then $a\in U_2HA$ for each $f\colon A\to B$ in $\mathcal{K}_1$" could be rewritten using $\alpha$ as "if $(U_1f)a\in \text{im}(\alpha_B)\subseteq U_1B$, then $a\in \text{im}(\alpha_A)\subseteq U_1A$ for all $f\colon A\to B$ in $\mathcal{K}_1$".