Definition. If $X, Y$ are two varieties, a morphism $\varphi: X \rightarrow Y$ is a continuous map such that for every open set $V \subseteq Y,$ and for every regular function $f: V \rightarrow k$, the function $\varphi^{*} := f \circ \varphi: \varphi^{-1}(V) \rightarrow k$ is regular.
let $O(Y)$ be the ring of all regular functions on $Y$.
Theorem .Let $X$ be a variety and $Y \subset \mathbb{A}^{n}$ be Affine variete and $\varphi: X \rightarrow Y$ be set function ( or any subset of $X$ send to a subset of $Y$) then: $\varphi$ is a morphism iff $\varphi^{*}$ send $O(Y)$ to $O(X)$ .
Is above theorem true when $Y$ be quasi-projective ?
No; the problem is that $O(Y)$ could be very small. For instance, if $Y$ is an irreducible projective variety, then $O(Y)$ will consist of only constant functions, so $\varphi^*$ will always send $O(Y)$ to $O(X)$ (since $O(X)$ always includes the constant functions), no matter what $\varphi$ is.