Let $S$ an arbitrary scheme and denote by $\Delta: S \to S \times S$ the diagonal immersion and $p_i: S \times S \to S$ the both projections to first resp second factor. (in following we will wlog talk only the projection on first factor, ie we set $p:=p_1$)
In general when $S$ isn't separated the image $\Delta(S) \subset S$ is only locally closed and not closed in $S \times S$. Since the restricted projection $p \vert _{\Delta(S)}: \Delta(S) \to S$ is an isomorphism that's not interesting to study this map. But the extended restriction
$$p \vert _{\overline{\Delta(U)}}:\overline{\Delta(U)} \subset U \times U \to U$$ looks more interessant. Which scheme theoretic map properties $\mathcal{Q}$ does $p \vert _{\overline{\Delta(U)}}$ inherit from $p \vert _{\Delta(S)}$? (ie $\mathcal{Q}$ stands for example beeing etale, smooth, flat etc)
Indeed the main motivation behind this question is the question if $p \vert _{\overline{\Delta(U)}}$ is always etale? (see This closely related question & discussion). Alexl gave the example where $S$ is the affine line with doubled origin and in this case it's quite obvious that $p \vert _{\overline{\Delta(U)}}$ is etale as well. But does this example cover all phenomena can occure by passing from $\Delta(S)$ to $\overline{\Delta(S)}$ as subscheme of $S \times S$?
More generally assume that $f: X \to Y$ a morphism of schemes and $S$ is locally closed subscheme of $X$ and $\overline{S}$ it's schematic closure in $X$. Assume the restriction $p \vert _S:S \to Y$ is etale (or has another fancy morphism property $\mathcal{Q}$ but I'm primary interested on etaleness as archetypical property), does the "extended" restriction $p \vert _{\overline{S}}:\overline{S} \to Y$ has also property $\mathcal{Q}$?
Is there any criterion / theory treating this problem and probably gives an answer for which morphism properties $\mathcal{Q}$ the above behaves well?
