Situation: I am trying to understand the proof of proposition 5.2 of Preframe Presentations Present by Johnstone and Vickers. The proposition states that the lower and upper power locale constructions commute.
Definitions: Let $D$ be a locale and write $\Omega D$ for the corresponding frame. The upper and lower power locale are defined via $$ \Omega P_U D = \operatorname{Fr}\langle \Box a \; (a \in \Omega D) \; \mid \Box \text{ preserves finite meets and directed joins} \rangle $$ and $$ \Omega P_U D = \operatorname{Fr}\langle \Diamond a \; (a \in \Omega D) \; \mid \Diamond \text{ preserves all joins} \rangle. $$
As a reminder: a frame homomorphism preserves finite meets and all joins, a sup-lattice is a poset with all joins and a sup-lattice homomorphism preserves all joins. Preserving all joins is equivalent to preserving both finite and directed joins.
In the proof: In order to show that $\Omega P_UP_L D = \Omega P_LP_UD$ the authors define maps $$ \Omega\theta : \Omega P_LP_UD \to \Omega P_UP_LD : \Diamond\Box a \mapsto \Box\Diamond a $$ and similar the other way round. Let's focus on the proof that $\Omega\theta$ is a frame homomorphism.
Question 1: The map $\Omega\theta$, the authors claim, must be equivalent with a sup-lattice morphism $\eta : \Omega P_U D \to \Omega P_UP_L D : \Box a \mapsto \Box\Diamond a$. Why is this true? (I can see that the generators of both $\Omega P_U D$ and $\Omega P_LP_UD$ correspond. Maybe we can view $\Omega\theta$ as the concatenation $\eta \circ \pi$, where $\pi : \Diamond\Box a \mapsto \Box a$, but would this be well-defined?)
Question 2: The authors proceed to show that $\Box a \mapsto \Box\Diamond a$ preserves directed joins, and then conclude that $\Omega\theta$ must be a frame homomorphism. Don't we need to show that $\Omega\theta$ also preserves finite meets and finite joins? (Or is this "easy"?)