  Let $X$ be a fixed projective scheme which is flat over a base scheme $S$.
The association taking a quasi-projective $S$-scheme $Y$ to the scheme
parametrizing $S$-morphisms from $X$ to $Y$ is functorial. We prove that this
functor preserves limits, and both open and closed immersions. As an
application, we determine a partition of schemes parametrizing rational curves
on the blow-ups of projective spaces at finitely many points. We compute the
dimensions of its components containing rational curves outside the exceptional
divisor and the ones strictly contained in it. Furthermore, we provide an upper
bound for the dimension of the irreducible components intersecting the
exceptional divisors properly.

Algebraic Geometry