# How to understand the definition of a final functor

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.

• Why wouldn't it be connected? There's a morphism connecting $id_A:A\to A$ to each of the other two objects in $(A\downarrow L)$. Mar 20, 2020 at 17:30
• 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!
– user761471
Mar 20, 2020 at 17:50
• 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?
– user761471
Mar 20, 2020 at 18:18
• 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. Mar 20, 2020 at 18:28
• Thank you (both), that has cleared everything up!
– user761471
Mar 20, 2020 at 19:57

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 $$P'$$.