From pg. 11 of Categories for the Working Mathematician:
Monoids. A monoid is a category with one object. Each monoid is thus determined by the set of all of its arrows, by the identity arrow, and by the rule for the composition of arrows. Since any two arrows have a composite, a monoid may then be described as a set $M$ with a binary operation $M \times M \rightarrow M$ which is asociative and has an identity (= unit).
Question: I do not see how we go from arrows $\rightarrow$ on a category with one object to a monoid $M$, which makes use of the notion of $\times$ (as described in the quote). So how exactly do categories with one object induce monoids (in the algebraic sense of the word)? Or, how do we move from $\rightarrow$ to $\times$ as expressed in the quote?