Monoids in Category Theory
I don't have a strong math background (engineering math) so I am at a bit of a disadvantage here but I have been trying to learn the broad strokes of Category Theory to help get a fuller picture of some of the Functional programming languages I use (Haskell, Scala, F#, etc.).
I am reading 'An Introduction to Category Theory' by Harold Simmons and ran into something in the first chapter regarding Monoids that leaves me a bit confused.
He states that a Monoid is a structure (R, *, 1) where R is a set, * is a binary operation on R and 1 is an element of R that functions as an identity element. So far, so good.
He goes on to state that a monoid morphism between two monoids:
p: R -> S
is a function that respects the structure of the monoids. He then explains that this means:
p(r*s) = p(r) * p(s), and p(1) = 1
where r and s are elements of R.
The question I have is that this seems to make certain assumptions: specifically, that R and S are the same monoid. I say this since p(r) is a morphism from R to S and * is defined for R, but not necessarily for S.
Have I missed something here?