Suppose that $X$ is a stack over the category of schemes which has the representable diagonal. If $X$ admits a surjection $f$ from a scheme $U$ which is also an open immersion (all pullbacks of $f$ are open immersions), then is $X$ a scheme?
For any $\phi : V \to X$ a morphism from a scheme, the pullback of $f$ is an isomorphism. I'd like to be able to conclude that $f$ is an isomorphism or equivalence of categories. Therefore, I am also asking the following question: if $f: X \to Y$ is a representable morphism of stacks, can the properties of $f$ to (isomorphism, equivalence, ...) be checked by base changing to some (sufficiently good / big) atlas $u : U \to Y$?
The moral question is - what properties of morphisms of stacks can be checked scheme-locally on the target?
I'm asking this because I'm trying to understand better the definition of Deligne-Mumford and Artin stacks, and it seems like asking the atlas to be an open immersion would be a reasonable way to strengthen the definition (and maybe get something uninteresting). (On the other end of the spectrum of Grothendieck topologies, what about asking for a f.p.p.f. atlas?)
I know the following: If $f : X \to Y$ is a morphism of schemes, and we know that for every affine subscheme $u : U \to Y$, the pullback of $f$ to $X \times U \to U$ is an isomorphism, then is $f$ an isomorphism. This is because inverses are unique and so can be glued together, but in the case of an equivalence of categories / with stacks, I am less sure.