It can be proven using Nakayama's lemma that if is a local Artinian $$-algebra, $_1$,$_2$ are finite type schemes flat over $$, and $:_1 \to _2$ is a morphism over $$ which restricts to an isomorphism on closed fibers, then is an isomorphism.
I am wondering if such a statement can be generalized to stacks in the following sense: Let $X, Y$ denote stacks over the category $\operatorname{Spec} A$, where $A$ is as above, and let $f: X \to Y$ be a morphism of stacks over $\operatorname{Spec} A$ which restricts to an isomorphism on $\operatorname{Spec} k$. Then is $f$ an isomorphism (equivalence of categories)?