Let $\mathbb D$ and $\mathbb E$ be two directed sets, then a map $f:\mathbb D\to \mathbb E$ is said to be final,if for any $e\in E$ there exists some $d\in D$ such that $f(d')\geq e$ whenever $d'\geq d.$
Question: For two arbitrary directed sets $\mathbb D$ and $\mathbb E,$ does there exists a third directed set $\mathbb A$ such that there exists two final maps $f_1:\mathbb A\to \mathbb D$ and $f_2:\mathbb A\to \mathbb E?$ Any help would be appreciated.
Let $\mathbb{A}$ be the family of finite subsets of $\mathbb{D}\cup \mathbb{E}$ ordered by inclusion. This is clearly a directed set. Let $f_1:\mathbb{A}\rightarrow \mathbb{D}$ be a function such that, for every finite set $F\in\mathbb{A}$, $f_1(F)\geq x$ for every $x\in F \cap \mathbb{D}$. This can be done using definable choice because $\mathbb{D}$ is directed.
Try to define $f_2$ analogously and prove that this works.