Morphism of algebraic stacks

In algebraic geometry, given algebraic stacks $$p: X \to C, \, q: Y \to C$$ over a base category C, a morphism $$f: X \to Y$$ of algebraic stacks is a functor such that $$q \circ f = p$$.

More generally, one can also consider a morphism between prestacks; (a stackification would be an example.)

Types
One particular important example is a presentation of a stack, which is widely used in the study of stacks.

An algebraic stack X is said to be smooth of dimension n - j if there is a smooth presentation $$U \to X$$ of relative dimension j for some smooth scheme U of dimension n. For example, if $$\operatorname{Vect}_n$$ denotes the moduli stack of rank-n vector bundles, then there is a presentation $$\operatorname{Spec}(k) \to \operatorname{Vect}_n$$ given by the trivial bundle $$\mathbb{A}^n_k$$ over $$\operatorname{Spec}(k)$$.

A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism.