In [comment](https://forum.azimuthproject.org/discussion/comment/18948/#Comment_18948) there are some ideas.
I would like to know the names of these ideas.

We have a monoidal skeleton-category
labeled \$$\mathbf{Matrix}_{skel} \$$
and two categories
\$$\mathbf{Matrix}\_{\mathbb{N}} \$$
and
\$$\mathbf{Matrix}\_{multiset}\$$ .

Here are example objects of those categories.

$\left( \begin{array}{cc} 1 & 1 \\\\ 1 & 0 \end{array} \right)$

and

$\left( \begin{array}{cc} \lbrace f \rbrace & \lbrace g \rbrace \\\\ \lbrace h \rbrace & \emptyset \end{array} \right)$

The two categories differ in the type of their cells and the definitions of their monoidal
operators but they are both similar to their skeleton in the same way.

What do we call these similarities?

@KeithEPeterson wanted to call them "matrix homomorphisms" but that is apparently wrong.

Edit: Thanks [Matthew Doty #81](https://forum.azimuthproject.org/discussion/comment/19020/#Comment_19020)
I corrected the latex and am studying what you posted.
That looks like what I was seeking.