Filtered vs Directed colimits

Question

I am having trouble with Adamek and Rosicky "Locally presentable and Accessible categories", specifically with the proof of theorem 1.5, namely

For every small filtered category $\mathcal D$ there exists a directed poset $\mathcal D_0$ and a cofinal functor $H \colon \mathcal D_0 \to \mathcal D$.

In the first part of the proof the authors assume that $\mathcal D$ has the property that every finite sub-category can be extended to a finite sub-category with a unique terminal object.

At some point the authors state

given two sub-categories $\mathcal A_1$ and $\mathcal A_2$ [sub-categories of $D$ with a unique terminal object] we extend $\mathcal A_1 \cup \mathcal A_2$ to a sub-category $\mathcal A$ with a unique terminal object.

Now my question:

how can we provide such extension $\mathcal A$.

In particular I have the following problem: given the two sub-categories $\mathcal A_1$ and $\mathcal A_2$ there is no reason why any sub-category containing both (and in particular the smallest sub-category containing them) should be finite, hence we cannot apply the hypothesis that allows to complete finite sub-categories to sub-categories with a unique terminal object.

Note the problem applies even if both $\mathcal A_1$ and $\mathcal A_2$ are finite.

Any help is appreciated.

Answer 1

I agree that this is a gap in the proof. An extremely careful proof that, in short, replaces finite subcategories with finite diagrams (allowing us to replace the problematic unions with disjoint unions) appears here:

H. Andréka, I. Németi, Direct limits and filtered colimits are strongly equivalent in all categories, Ban. Cen. Pub. 9 (1982), 75-88.

Thanks to @user12580 for digging up the reference.