What does the endofunctor/monad that sends a set to the set of finite words on the set do to morphisms? Suppose we have a monad $T:Set \rightarrow Set$ that sends a set X to the set of finite words on the set X, with the unit and multiplication being inclusion and concatenation respectively. What does such a monad do to a function $f: X \rightarrow Y$ for two sets $X$ and $Y$? 
I assume it defines the image pointwise, i.e. $T(f)(xy) = f(x)f(y)$, however I cannot find a source confirming this. I suppose it is a question of convention, and if so then I am just looking for the conventional answer. I assume there are other (not so sensible) monads, however I am not interested in these.
If this is the case, then I think problems will arise when the domain contains composite words. Consider a function $f: \{x,y,xy\} \rightarrow \{x,y\}$, where $xy$ is the word formed by the concatenation of the words $x$ and $y$. Then what is $T(f)$? If it is defined pointwise, then we would have to have a coherence $f(xy) = f(x)f(y)$ in order for the pointwise definition to make sense.
If anyone could offer any clarification on this matter (or sources discussing it) it would be much appreciated.
It is possible that my understanding of the set image of the functor is incorrect. I am currently working under the assumption that $T^2(X) = T(X)$, however the more I think about this the more silly this seems. This is probably the root of my problem.
 A: You seem to be referring to the free monoid monad. The underlying functor $T\colon \mathbf {Set}\to \mathbf{Set}$ of this monad is the free monoid functor which sends a set $S$ to the free monoid on $S$, namely all finite words in the alphabet $S$. This functor is defined on morphisms in the pointwise manner you suggest, namely for a function $f\colon S\to T$ and a word $w=s_1\dots s_n$ in $T(S)$ we have $T(f)(w)=f(s_1)\dots f(s_n)$. This is not ambiguous though since in the free monoid $T(S)$ every word decomposes uniquely as a concatenation of letters. In the example you are giving you need to treat $xy$ as a single symbol. Your set $S=\{x,y,xy\}$ is simply a set with three things in it. The set does not 'know' (nor does it care) that to you it seems like one of these things is already a concatenation of the other two things. 
You can, of course, consider the monoid generated by $\{x,y,z\}$ subject to the relation $z=xy$. But this isn't a free monoid (but it is a quotient of one, namely the free monoid on three elements).   
