The “if and only if” version is meant for talking about short exact sequences $0 \to M \to M' \to M'' \to 0$. E.g. as implied by lemma 12.10.2 of the stacks project page.

]]>If M=M’, the map M→M’ is identity, and the map M’→M” is the zero map, then the sequence M→M’→M” is exact.

Thus, the “only if” part of “if and only if” implies that *any* object M” belongs to any Serre subcategory.

So it seems like the “only if” part should be removed.

]]>Please note that the $n$Lab page *Serre subcategory* is at best a stub that is hardly meant to be authorative but is instead waiting for a kind expert soul to take care of it!

The page history shows that its content was jotted down 12 years ago (rev 1) by a user no longer active here, and not substantially touched by any expert author. Related problems with related entries had recently been raised in comment 94319 and I have tried to patch it up then (comment 94328), but, as I disclaimed there, I am neither expert on this nor am I investing serious thoughts into it (being busy elsewhere).

So if there is a contradiction between the StacksProject and these pages, it’s certainly these pages that need attention. You would do the $n$Lab community a great service if you took care of this! (To start with, this may require minimal work, just replacing wrong or non-existent definition with correct ones, or even just with pointers to correct ones). Editing pages here is pretty straightforward, and I’d be happy to help if any issues arise.

]]>Recently, there was a discussion in the page about Serre subcategories in The Stacks project. I usually follow both nLab and The Stack project for these things but it seems that the definitions of Serre subcategories are different here and there. I just want to clarify if both definitions are correct or there is an error on any of them. Am I missing something? The discussion is about the use of “if” or “iff” conditions under the arrows involving $M,M',M''$ (or $A,B,C$) in the pages Serre subcategory and https://stacks.math.columbia.edu/tag/02MO (there is even more discussion (going on) in https://stacks.math.columbia.edu/tag/02MN).

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