I feel like there's something obvious I'm missing here, and I'm not looking for a whole answer, but rather just a pointer in the right direction.
Suppose you have the projection from a point $\mathbb{P}^n \backslash \{P\} \to \mathbb{P}^{n-1}$, where $P = (0:...:0:1)$. This is not defined at $P$, but the claim is that it extends to a morphism from $B_P(\mathbb{P}^n)$ (blow up at $P$) to $\mathbb{P}^{n-1}$. I'm trying to figure out exactly how to define this morphism.
Since the blow up just replaces $P$ with a copy of $\mathbb{P}^{n-1}$, it seems like the most natural morphism to consider should just be the identity, specifically from $\{P\} \times \mathbb{P}^{n-1} \to \mathbb{P}^{n-1}$. This seems to make sense; the definition of morphism $\varphi$ between quasi-projective varieties $X \to Y$ that I have is a Zariski continuous map such that if $f$ is regular on an open set $U \subset Y$, then $f \circ \varphi$ is regular on $\varphi^{-1}(U)$, and as far as I can tell, this new map should satisfy these conditions, since I'm basically giving two morphisms on two disjoint parts of the blow-up.
Does this work? And in this vein, could I use any morphism from $\{P\} \times \mathbb{P}^{n-1} \cong \mathbb{P}^{n-1} \to \mathbb{P}^{n-1}$?