For example, consider a pair of monoids of degree three. That is, in each monoid there are three arrows $a_1$, $a_2$ and $a_3$, all of which are reflexive. Would it make sense to identify the composition of $a_1$ with $a_2$ with only one of the arrows? If so, then there could be several distinct monoids of degree three, for example one with $a_1 *a_2$=$a_3$ and another with $a_1*a_2=a_1$ but $a_1*a_2\ne a_3$, etc.. On the other hand, if the composition of $a_1$ with $a_2$ is always identified with all three arrows then all monoids with degree three would represent the same category.
So my question is about definitions - what defines a category, the specific arrows resulting from composition or the pairs of elements that are connected by composition of arrows?