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This is a theorem in Locally presentable and accessible categories, by Adamek, Rosicky.

For every (small) filtered category $D$ there exists a (small) directed poset $D_0$ and a cofinal functor $H:D_0 \rightarrow D$.

It says as an example to illustrate the difficulty, consider

$D$ with one object $d$ two morphisms $id$ and $f=f\cdot f$. Then $D_0$ has to be an infintie directed category $$d \xrightarrow{f} d \xrightarrow{f} d \cdots $$

There must be some serious misunderstanding in my knowledge of a cofinal functor. According to the book, we require

Given $f:d \rightarrow Hd_0, f':d \rightarrow Hd_1$, there exists $g:d_0 \rightarrow d , g': d_1 \rightarrow d'$, such that $$ Hg \cdot f = Hg' f' $$ where $d \in D, d_0, d_1, d' \in D_0$.

How could this be attaiened by the example?

Simply consider $$ d \xrightarrow{id} Hd=d, d \xrightarrow{f} Hd=d$$ Then there is no way to find such $g,g'$.

What am I missing?

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Take any $d'$ such that $d_0,d_1 < d'$ in the poset $D_0$. Then the arrows $g,g'$ are not identities, so that $H(g)=f=H(g')$, and thus $H(g)\circ id=f=f\circ f=H(g')\circ f$.

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  • $\begingroup$ Ah, I thought $f^2=id$... Thanks a lot. $\endgroup$ – Bryan Shih Aug 8 '19 at 8:03

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