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 $\mathcal D$ 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 thr 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.