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I left the following comment on Pullbacks in this category of vector spaces are intersections:

Are the morphisms in your category isometries? Otherwise, I'm not sure why the category is left cancellative. If so, then it's surely more or less diagram chasing that the pullback of $\require{AMScd}\begin{CD} V' @>f'>> V \end{CD}$ and $\begin{CD} V'' @>f''>> V \end{CD}$ is $$\begin{CD} f'(V') \cap f''(V'') @>f^{\prime\,{-1}}>> V' \\ @Vf^{\prime\prime\,{-1}}VV @VVf'V \\ V'' @>f''>> V. \end{CD}$$

Are the morphisms in your category isometries?  Otherwise, I'm not sure why the category is left cancellative.  If so, then it's surely more or less diagram chasing that the pullback of $\require{AMScd}\begin{CD} V' @>f'>> V \end{CD}$ and $\begin{CD} V'' @>f''>> V \end{CD}$ is $$\begin{CD} f'(V') \cap f''(V'') @>f^{\prime\,{-1}}>> V' \\ @Vf^{\prime\prime\,{-1}}VV @VVf'V \\ V'' @>f''>> V. \end{CD}$$

Then I realised that my notation wasn't great, and it should really be:

Are the morphisms in your category isometries? Otherwise, I'm not sure why the category is left cancellative. If so, then it's surely more or less diagram chasing that the pullback of $$\require{AMScd}\begin{CD} V' \\ @. @VVf'V \\ V'' @>f''>> V. \end{CD}$$ is $$\begin{CD} f'(V') \cap f''(V'') @>f^{\prime\,{-1}}>> V' \\ @Vf^{\prime\prime\,{-1}}VV @VVf'V \\ V'' @>f''>> V. \end{CD}$$

Are the morphisms in your category isometries?  Otherwise, I'm not sure why the category is left cancellative.  If so, then it's surely more or less diagram chasing that the pullback of $$\require{AMScd}\begin{CD} V' \\ @. @VVf'V \\ V'' @>f''>> V. \end{CD}$$ is $$\begin{CD} f'(V') \cap f''(V'') @>f^{\prime\,{-1}}>> V' \\ @Vf^{\prime\prime\,{-1}}VV @VVf'V \\ V'' @>f''>> V. \end{CD}$$

(I now see that I put that $V'$ in the first row in the wrong place, but that's not the point.) However, one of the @s—I'm not sure which one—was interpreted as an @-notification:

Only one additional @user can be notified; the post owner will always be notified

Bonus question: Although the above all seems to show up correctly on posting, it messes up on preview: what shows up on the main site as

Rendered comment on main site

appears in the preview of the initial post (but not on the preview of the edit I am doing now) as

Rendered preview of the same markup on MMO

(although it appears to be fine on post).

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    $\begingroup$ Regarding the bonus question, it might be a weird Markdown/MathJax interaction, where first the Markdown processor treats two backslashes as one escaped backslash, so not MathJax can't find the linebreak points. It happens occasionally to me, especially if my internet connection is unstable. There are some related posts on Meta SE, and I'm sure also on Math SE, but no real fix. $\endgroup$ Commented May 9, 2023 at 22:11
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    $\begingroup$ Both versions of the comment throw the same error when I try to post them, yet you were obviously able to post the first version. Not quite sure what's going on. In fact, any comment that I write with two arbitrarily placed @ symbols throws the same error, so it might not have anything to do with AMScd per se. $\endgroup$ Commented May 9, 2023 at 22:27
  • $\begingroup$ Are the morphisms in your category isometries? Otherwise, I'm not sure why the category is left cancellative. If so, then it's surely more or less diagram chasing that the pullback of $\require{AMScd}\begin{CD} V' @>f'>> V \end{CD}$ and $\begin{CD} V'' @>f''>> V \end{CD}$ is $$\begin{CD} f'(V') \cap f''(V'') @>f^{\prime\,{-1}}>> V' \\ @Vf^{\prime\prime\,{-1}}VV @VVf'V \\ V'' @>f''>> V. \end{CD}$$ $\endgroup$
    – LSpice
    Commented May 9, 2023 at 23:24
  • $\begingroup$ @TheAmplitwist, re, the first comment doesn't post even if you copy and paste it exactly as written? I tried here to see if it was some temporary glitch, and it posted without trouble. I can't imagine why it would post for me and not for you …. (Well, on this post, because I'm the author, maybe, but that wasn't the case on the main-site post.) $\endgroup$
    – LSpice
    Commented May 9, 2023 at 23:25
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    $\begingroup$ Ah, I was able to post the first comment just now on my MacBook. I had earlier tried and failed to post via my iPhone. Perhaps something went wrong in copying the right comment on the iPhone's small screen. $\endgroup$ Commented May 10, 2023 at 5:14

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