Morphism of algebraic stacks

In algebraic geometry, given algebraic stacks formula_1 over a base category "C", a morphism formula_2 of algebraic stacks is a functor such that formula_3.
More generally, one can also consider a morphism between prestacks; for this, see prestack#Morphisms (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 formula_4 of relative dimension "j" for some smooth scheme "U" of dimension "n". For example, if formula_5 denotes the moduli stack of rank-"n" vector bundles, then there is a presentation formula_6 given by the trivial bundle formula_7 over formula_8.
A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism.