Mapping stacks and categorical notions of properness

Understand

One fundamental consequence of a scheme $X$ being proper is that the functor classifying maps from $X$ to any other suitably nice scheme or algebraic stack is representable by an algebraic stack.

• This result has been generalized by replacing $X$ with a proper algebraic stack.
• We show, however, that it also holds when $X$ is replaced by many examples of algebraic stacks which are not proper, including many global quotient stacks.
• This leads us to revisit the definition of properness for stacks.