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:

**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

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

(although it appears to be fine on post).

`@`

symbols throws the same error, so it might not have anything to do with`AMScd`

per se. $\endgroup$