Suppose $X$ and $Y$ are projective varieties and $\Gamma \subset X \times Y$ is a graph, that is for any $x \in X$ we require $\pi_{Y}(\pi^{-1}_{X}(x))$ to be a point in $Y$, call it $f(x)$. Does $\Gamma$ define a morphism $f:X \to Y$?
Clearly if $X$ is normal then since $\pi_{X}:\Gamma \to X$ is bijective it must be an isomorphism and we have $f=\pi_{Y} \circ \pi_{y}^{-1}: X \to Y$ as required.
Can we still define a morphism in a similar fashion when $X$ is not normal? $Y$ can be smooth if that makes a difference.