[mikorym]

So yes, this is equivalent to FinVect of the field of which entries in the matrices consist.

The difference is that here you construct the category from a simpler premise. To construct FinVect you need to include all set objects with structure satisfying some axioms.

The category of matrices is simply positive integers with as morphisms n x m matrices between the two integers. Composition is matrix multiplication.

Here [1] is a nice overview. If you can follow what is going on there, it is worth while looking at II, III and IV.

[1] https://unapologetic.wordpress.com/2008/06/02/the-category-o...