I'm trying to wrap my head around sheafification via the étale space construction from, say, Hartshorne's or Liu's books on algebraic geometry (I guess this is a construction of Bourbaki?). In these constructions, for a presheaf $\mathcal{F}$ on a topological space $X$ with stalks $\mathcal{F}_x$ at $x$, one defines the sheafification $\mathcal{F}^{\dagger}$ to have sections $$\mathcal{F}^{\dagger}(U) = \left\{f: U \to \amalg_{x \in U} \mathcal{F}_x\, |\, \star \right\}$$ where $U \subseteq X$ is open and $\star$ is the condition that for each $x \in U$ there exists an open neighborhood $x \in V \subseteq U$ and a section $s \in \mathcal{F}(V)$ verifying $f(y) = s_y$ for all $y \in V$. That is, that $f$ doesn't pick germs arbitrarily, but picks out "compatible local sections".
In thinking of how I would've intuited a definition of $\mathcal{F}^{\dagger}(U)$ as a limit over some diagram of the $\mathcal{F}_x$'s, this notion in terms of functions struck me as odd until I realized that, in Set at least, $$\prod_{x \in U} \mathcal{F}_x \cong \left\{f: U \to \coprod_{x \in U} \mathcal{F}_x \right\}.$$
Q1. This implies that I can construct sections of a sheaf, not just here but in general as some kind of limit of the stalks, but I'm unsure what the transition maps should be. I am familiar with a construction of the sections over an open $U$ relative to an open cover, but I'm not sure how to take that "to the stalks". How can one construct $\mathbf{\mathcal{F}^{\dagger}(U)}$ as a limit over a diagram of stalks and homomorphisms?