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Borceux (Handbook of Categorical Algebra, Volume 1, Definition 2.11.1) defines final functors as those functors for which restriction preserves limits, which is the exact opposite of what the nLab says (i.e., restriction along final functors preserves colimits).
Is this just Borceux’s idiosyncrasy? His books generally appear to use conventional terminology. Shouldn’t this be mentioned in the article?
The new version is confusing: “final” appears as a name for two opposite notions!
Maybe we should offer some guide to the terminology. I already mentioned Borceux’s book terminology: cofinal for colimits, final for limits. Mac Lane’s terminology: final for colimits, initial for limits. What other canonical sources should be considered?
I think Borceaux is the odd one out. We can mention his terminology, but we shouldn’t suggest that its use be continued.
I think in the definition of final functor, there is a typo: “$F : C \to D$ is final if for every object $d \in D$ the comma category $(d/F)$ is……” should be: $F : C \to D$ is final if for every object $c \in C$ the comma category $(c/F)$ is……
Jishen Du
$(d/F)$ is correct; $(c/F)$ doesn’t type check. In particular, $(d/F)$ is the comma category from the functor $1 \to D$ that picks out $d$ to the functor $F : C \to D$.
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