Example : Consider the (open, not compactified) moduli space of stable maps $ \mathcal M\_g(1,d)$ of maps of smooth curves of genus $ g$ to $\mathbb P^1$. To each map
we associate its branch divisor, which is an element of $ Sym^r(\mathbb P^1)$. Then we should have a morphism from $ \mathcal M\_g(1,d)$ to $ Sym^r(\mathbb P^1)$. How do
we prove this?


In general, if we have a family of objects $A$ and for each $ a \in A$ we can
choose an element $ b \in B$ that "depends continuously" on $ a$. How do we prove
that we have established a morphism $A \to B$? What is the general method to do this?


Note about the example: Extending the morphism to the Kontsevich compactification of the moduli space
was the main objective of a paper by Fantechi-Pandharipande. But I couldn't filter out
the proof that the map is a morphism, or the proof wasn't there. 



Proving that a map is a morphism
It might help to identify $Sym^r(\mathbb P^1)$ with the Hilbert scheme of degree $r$ effective divisors on $\mathbb P^1$. The Hilbert scheme represents a functor, as does the space $\mathcal M\_g(1,d)$, and so you can (try to) construct a map from one to the other by thinking in terms of the functors they represent.


As to why $Sym^r(C)$ is the Hilbert scheme of degree $r$ effective divisors when $C$ is a smooth projective curve, this is a (non-trivial, it has always seemed to me) exercise in making contact with reality from the somewhat more abstract world of flat families.


