Can the Diagram in the Article Be Interpreted as Commutative?

[Stephen Tashi]

TL;DR Summary

Does the given example of commutative diagram use conventional notation?

I'm used to seeing commutative diagrams where the vertices are mathematical objects and the edges (arrows) are mappings between them. Can the diagram ( from the interesting article https://people.reed.edu/~jerry/332/25jordan.pdf ) in the attached photo be interpreted that way?

In the article:

$k[x]$ is the ring of polynomials over a field k.

$V$ is a vector space.

$T$ is a linear transformation on $V$

I understand $T$ and $X$ as maps, but do the vertical arrows go from a map to the argument of a map?

 

[fresh_42]

The arrows (all of them, vertical and horizontal) go from element to element. Their LaTeX code is \mapsto.

 

[Stephen Tashi]

I'll understand the vertical arrow on the left this way: There is an (unnamed) isomorphism mapping $V$ to a direct sum of quotient modules. So $g(x) + <f_i(X)>$ is one element of that direct sum and it is in the coset of the $<f_i(X)>$. So the unnamed isomorphism maps an element of the direct sum to a vector $v$ in $V$.

 

[mathwonk]

the visual clue to what fresh and Stephen are saying is that the arrows have little tails at the beginning which are perpendicular to the arrow.