Let be the category for matrices such that , and . We can define a composition function using knowledge of matrix multiplication:

where .

Now let , , and in that . From the composition of matrices, we can now imply that the category is associative:

.

Here is the communative diagram for the associativity:

Therefore we may show that the identity where and :

.

We have defined a category of matrices. (hopefully…)

*Note: Please let me know of any errors I have made as I want to be able to correct them as I learn the concepts.*

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Just write $\hom(a,b)\times\hom(b,c)\to\hom(a,c)$ to keep things simple – there’s no need to use subscripts with three things. Note that I repeated the letter $b$ twice, instead of writing four letters $a,b,c,d$. In order to compose to morphisms, the codomain of the one must be the domain of the other. A similar thing must be said for your commutative diagram and all the different natural numbers you use. What does $f:\hom$ mean by the way? Why did you replace the letters $f,g,h$ with $\hom$-sets in your symbolic derivation? That makes no sense at all to me. If you had just said there is a category whose objects were natural numbers and whose hom-sets $\hom(m,n)$ were $m\times n$ matrices over a field $F$, because matrix multiplication is associative and the identity matrix acts as the identity, that single sentence would have been enough for me to express everything you intend to express here.

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Ah. I see. I believe that I just made it a whole lot harder for myself by writing everything out and assigning separate letters of make everything even more complicated. $f:\Hom$ is just a morphism that I defined to “make things easier for the reader.” But I guess not. The symbolic derivation you mean in the communative diagram? I was just showing associativity. Or am I referring to something different than to what you are thinking of? I understand the rest of what you said. I just need to learn how to simplify my explanations and not make it more complicated than it already is.

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