Fix some algebraically closed field k. Consider the morphims $\psi, \phi:A^1 \rightarrow A^1\sqcup{0'}$ where $\phi(k):=k^-1$ if k is different from zero and $\phi(k):=0'$. Likewise define $\psi$ as above but instead in the first case it assigns a non-zero point itself (instead of it's inverse).
The first case defines an object isomorphic to $P^1$, in the category of varieties but the second defines a pre-variety which is not a variety. Why is this?
P.S.: by variety X I mean a pre-variety who's diagonal is closed in $X\times X$.
Thank in advance guys and gals :)