What does this line in Lang's "Algebra" mean? Lang's "Algebra" says the following:

Let $S$ be a set. A mapping $S\times S\to S$ is sometimes called a law of composition (of $S$ into itself).

I always thought $S\times S\to S$ implied a binary operation on two elements of $S$, and $S$ being closed on that binary operation. 
I also thought the word "composition" belonged to the world of mappings.
I don't see how a binary operation can be called a mapping.   
EDIT: I must confess I have come across such a usage of the term "composition" before, but could never quite get the motivation behind it. Really hoping for an elaborate answer to shed light on this issue; something that I am sure confuses other autodidacts out there too. 

The first groups that were studied are symmetry groups, like
$$
S_n = \{\, \varphi: [n]\to[n] \,\mid\, \text{$\varphi$ bijective}\,\},\quad
\text{where $[n]=\{1,2,\dots,n\}$}
$$
or the symmetry groups of geometric objects like polyhedra. In all of these, the elements actually are self-maps on some set and the group composition is the actual composition of these maps as maps.
These examples motivate why we sometimes call the operation in a group a composition.
