Are morphisms from affine schemes to Artin stacks affine morphisms?

Question

It is explained in this MO discussion that if one has a morphism $f:X\rightarrow Y$ of schemes such that $X$ is an affine scheme, then $f$ need not be an affine morphism. However, if $Y$ is separated, then $f$ is indeed affine.

Question: what happens is if $Y$ is an Artin stack? (But $X$ is still an affine scheme). What are some natural conditions on $Y$ to ensure that $f$ is an affine map?

The example I have in mind is when $Y$ is the stack of rank $n$ vector bundles.

Answer 1

The obvious natural condition is that the diagonal $\Delta_Y:Y\to Y\times Y$ is affine. This holds for stacks of vector bundles: it just means that if $E$ and $F$ are rank $n$ vector bundles on $X$, then $\underline{\mathrm{Isom}(}E,F)$ is (representable and) affine over $X$; it is in fact a $\mathrm{GL(E)}$-torsor.