If $X$ and $Y$ are topological spaces, then $X^Y$ is the set of all continuous functions from $Y$ into $X$.
There is some text in my textbook that leads me to believe this is different from $Hom(Y,X)$.
But in the category of topological spaces, isn't $Hom(Y,X)$ just the set of all continuous functions $Y \to X$? So what is the difference?
Here is a 'snippet' of the text that has lead me to the conclusion - (I can write out more if it helps)
it follows that each $F_z$ is continuous and that the target of $F^\#$ is indeed $X^Y$ (not merely $Hom(Y,X))$