I have been following the proof of Theorem 2.2 in Peter Scholze's Lectures on Condensed Mathematics, and have had a hard time following some steps:
The setup is that there are projective objects $T$ in a category of sheaves (of abelian groups over extremally disconnected spaces) and we want to show that, for any object $M$, taking the coproduct of all such $T$ which have arrows $T\rightarrow M$ yields a surjection $\bigoplus T\rightarrow M$.
The author uses Zorn's lemma to get "a maximal subobject $M'$ of $M$ admitting such a surjection". I am struggling to see what poset $M'$ is a maximal element of, since $M$ should be the maximal element of the poset of subobjects of $M$. Is this supposed to be seen as a "subposet" of the poset of subobjects of $M$ constituted only by subobjects with a surjection $X\rightarrow M$? Because then I can't see why every chain there would have an upper bound, in order to apply Zorn's lemma. Does the image of the induced map $\bigoplus M'\rightarrow M$ work?
Further on, he states that, given a maximal such $M'$, if $M/M'$ was not 0, then the there would be a non-zero map $T\rightarrow M/M'$ for some $T$ (this I understand and is concluded by previous details I didn't provide) which would then factor through $M\rightarrow M/M'$ since $T$ is projective. Then he claims that this contradicts the maximality of $M'$, but I just cannot see why.
Thanks in advance for any clarifications. The scenario in the proof is a bit more specific, so I can provide more details if needed, but I feel like it shouldn't be anything fancy and I'm just confused about the maps in question.