I'm working through the book Algebraic Set Theory by Joyal & Moerdijk. However, I stumbled upon some contruction which I did not manage to understand yet. In the proof of Proposition 4.2, we define a map $\bar{g}$ on the way it looks when composed with another map. I do not understand how this defined the map $\bar{g}$ itself. One thing I could think of was that it is defined on generalized elements in $P_s(X)$, and by this way should define the whole of the map, but I am not sure how this formally should look like. This is the proof. Does anyone have some insights on this?
Thanks in advance
Added later: I'll try to give some extra context. We assume our category to be a Heyting pre-topos. We work with a class of small maps, which is a certain class of maps satisfying a lot of axioms. We write $\mathcal{S}$ for this class. The lattice being $S$-complete means that all suprema along small maps exist. The object $P_{\mathcal{s}}(X)$ is the object representing all small families of subobjects of $X$. It might be that the situation is too specific to give an answer without knowledge of the broader context.
It seems to me that we're given a set $X$ and we form the power set $P_s(X)$ which is a sup-lattice (true enough) and if $g$ is a map into a complete sup-lattice $L$ we can extend it to $P_s(X)$ because on singletons $\{x\}$ $(x \in X)$ we define $\overline{g}((\{x\})= g(x)$ (no choice because of the commutativity) and if $A \in P_s(X)$, so $A \subseteq X$, we define $g(A)= g(\bigvee_{x \in A} \{x\})$ by $\bigvee_{x \in A} g(x)$, using the completeness of $L$. I think this is what it comes down to (minus the jargon about smallness and projections). But maybe someone who knows the book can shed a better light.. This is how it would work in the category Set, and this looks like an abstract version of that.