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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.

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  • $\begingroup$ If $T_1$ and $T_2$ are the terminal objects, use the fact $\mathcal{D}$ is filtered to prove the existence of extra objects and arrows you can adjoin to $\mathcal{A}_1 \cup \mathcal{A}_2$ as needed. E.g. if $T_1$ and $T_2$ are distinct, then you can ask for an object $X$ and arrows $T_1 \to X$ and $T_2 \to X$. I presume that every case will be straightforward; the problem is simply enumerating all of the cases you need to check. $\endgroup$ – user14972 Aug 18 '18 at 15:11
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    $\begingroup$ @ArnaudD. the union of two sub-categories is not a category and the smallest category containing them is not necessarily finite: consider the graph with two points and two arrows connecting them, the free category generated by them contain two finite sub-categories finite (those generated by the two arrows of the graph) but the category generated by their union is the whole free category which is not finite. $\endgroup$ – Giorgio Mossa Aug 18 '18 at 17:07
  • $\begingroup$ @Hurkyl I think the problem is more complex than that: the problem is that the smallest sub-category containing $\mathcal A_1 \cup \mathcal A_2$ could be infinite so even if we add the object $X$ as you say there is no easy way to build a weak-coequalizer for the morphisms of this biggest sub-category, at least in the case it has infinite morphisms, which I think the example in my previous comment should prove could be the case. $\endgroup$ – Giorgio Mossa Aug 18 '18 at 17:12
  • $\begingroup$ @giorgiomossa Right, I had a disjoint union in mind, sorry. $\endgroup$ – Arnaud D. Aug 18 '18 at 21:24
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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: http://matwbn.icm.edu.pl/ksiazki/bcp/bcp9/bcp919.pdf. Thanks to @user12580 for digging up the reference.

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