tikz-cd arrow between arrows is asymmetric (too short in one direction)

I want to draw an arrow (A) between two arrows (B and C) in tikz-cd, so that A is directly touching both B and C, like in the following diagram.

The code below seems like the sensible attempt at this diagram, but it’s asymmetric: the middle arrow does not touch the lower arrow.

documentclass{article}
usepackage{tikz-cd}

begin{document}

begin{tikzcd}
    bullet && bullet
    arrow[""{name=0}, from=1-1, to=1-3, bend left]
    arrow[""{name=1}, from=1-1, to=1-3, bend right]
    arrow[from=0, to=1]
end{tikzcd}

begin{tikzcd}
    bullet && bullet
    arrow[""{name=0}, from=1-1, to=1-3, bend left]
    arrow[""{name=1}, from=1-1, to=1-3, bend right]
    arrow[from=1, to=0]
end{tikzcd}

end{document}

This is the result:

(The example on the bottom has the middle arrow in the opposite direction for comparison. It demonstrates that it is the lower arrow that is the issue, rather than the target arrow.)

Why is this happening, and how can I fix it?

Note that I do not want to simply shorten the middle arrow with a negative length: the solution must work with any pair of arrows, and I don’t want to have to guess how much to shorten the middle arrow by. (Also, in some cases I want to shorten the middle arrow symmetrically, which is awkward if one direction has to have extra adjustment applied to it.)

documentclass{article}
usepackage{tikz-cd}

begin{document}

begin{tikzcd}
    bullet && bullet
    arrow[""{name=0}, from=1-1, to=1-3, bend left]
    arrow[""{name=1, below}, from=1-1, to=1-3, bend right]
    arrow[from=0, to=1]
end{tikzcd}

begin{tikzcd}
    bullet && bullet
    arrow[""{name=0}, from=1-1, to=1-3, bend left]
    arrow[""{name=1, below}, from=1-1, to=1-3, bend right]
    arrow[from=1, to=0]
end{tikzcd}

end{document}