$\require{AMScd}$Teaching coend calculus to a PhD student led me to this "elementary" computation that I would like to perform explicitly.

Consider the functor $F : (\mathbb N,\le)^\text{op}\times (\mathbb N,\le) \to \mathrm{Vect}$ sending a pair of natural numbers $(n,m)$ to the vector space $\hom(\mathbb R^n,\mathbb R^m)$ (so: choosing a basis, real valued $m\times n$ matrices).

$F$ acts on inequalities $p\le q$ by suitable pre- and post-composition with the inclusions $i_{p\le q} : \mathbb R^p \to \mathbb R^q$ "padding $q-p$ zeroes at the end of the vector $(a_1,\dotsc,a_p)$".

What is the coend of $F$?

An instructive unraveling of the definition tells us that we have to compute the cokernel of the map $$ \begin{CD} \bigoplus_{n\le m} M_{m,n}(\mathbb R) @>>> \bigoplus_p M_p(\mathbb R) \end{CD} $$ defined at the component $n\le m$ sending a linear function $H : \mathbb R^m\to\mathbb R^n$ to $H\circ i - i\circ H$ (this is slightly imprecise, but the abuse of notation is harmless: both compositions are defined, and the difference is taken padding the smaller matrix with zeroes until it reaches the same dimension of the other). Up to this padding operation, this is a sort of commutator of $H$ with the inclusion $i_{n\le m}$.

The vector space I am interested now arises by killing the image of this commutator map in $\bigoplus_p M_p(\mathbb R)$; I would like to understand this quotient in terms of some concrete representation (for example regarding said cokernel inside the $\mathbb R$-algebra of all linear operators $\mathbb R^{\mathbb N} \to \mathbb R^{\mathbb N}$), but at the moment I can only gather a number of sparse, inconclusive observations. For example:

- I believe there is a "simpler" expression for $\int^n F(n,n)$ in terms of a filtered colimit of cokernels of the partial maps $\bigoplus_{n\le m} M_{mn}(\mathbb R) \to \bigoplus_p M_p(\mathbb R)$, because they factor through $\bigoplus_{p=1,\dotsc,m} M_p(\mathbb R)$ and have a cokernel $Q_m$ in a chain $Q_1\to Q_2\to Q_3\to\dotsb$.
- $\bigoplus_p M_p(\mathbb R)$ seems to be a filtered $\mathbb R$-algebra (given two matrices, pad zeroes to let the dimension match, and perform operations there). Is it the case then that $\int^n F(n,n)$ inherit some filtration and compatible algebra structure?

I think some advanced linear algebra is enough to answer the question, so please, don't be scared if you're not familiar with coends or even with category theory.