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I am struggling to understand the definition of a final functor given on page 217 of MacLane's "Categories for the Working Mathematician." It says that a functor $L \colon J' \rightarrow J$ is final if for each $k \in J$ the comma category $(k/L)$ is nonempty and connected. Let's suppose that $L$ is an inclusion, and $J$ is the pushout data $$ B \leftarrow A \rightarrow C $$ If $L$ is simply the identity, it does not seem to satisfy the conditions of being final. That is, the comma category under the $A$ is not connected as far as I can tell. But surely the identity functor is final?

I'm sure I am misunderstanding something and would appreciate if someone could set me straight.

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  • $\begingroup$ Why wouldn't it be connected? There's a morphism connecting $id_A:A\to A$ to each of the other two objects in $(A\downarrow L)$. $\endgroup$ Mar 20, 2020 at 17:30
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    $\begingroup$ Wow, do I feel silly...yes, I see it now. I had it in my head that you'd need a morphism from $B$ to $C$ or the other way around. Total brainfart. Thank you for clarifying! $\endgroup$
    – user761471
    Mar 20, 2020 at 17:50
  • $\begingroup$ A follow-up question as well. Why would the inclusion of the category with just the objects $B \ A \ C$ (removing the two original morphisms) not be final? $\endgroup$
    – user761471
    Mar 20, 2020 at 18:18
  • $\begingroup$ If you take $J'$ to be the category with the same objects but only the identity morphisms, then $A/L$ also has only identity morphisms, but it has three objects $A\to A$, $A\to B$, $A\to C$ so it is not connected. $\endgroup$
    – Arnaud D.
    Mar 20, 2020 at 18:28
  • $\begingroup$ Thank you (both), that has cleared everything up! $\endgroup$
    – user761471
    Mar 20, 2020 at 19:57

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One way i build intuition is from when considering colimits indexed by directed posets.

If $P$ is some directed poset and $P'\subseteq P$ is some subset. Then the inclusion $i:P'\to P$ is a final functor if for all $p \in P$ there exists some $p'\in P'$ such that $p\leq p'$. Hence it is sufficient to compute colimits indexed by $P'$.

This plays nice with my intuition since in nice situations such colimits should be fought of as unions in which thise should be very clear.

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