Consider the fibred category p : *Spaces* —> *(Sch)* where an object of *Spaces* over the scheme U is an algebraic space X over U. A morphism (f, g) : X/U \to Y/V is given by morphisms f : X —> Y and g : U —> V fitting into an obvious commutative diagram.

Theorem: This is a stack over (Sch)_{fppf}.

In essence the thing you have to prove here is that any descent data for spaces relative to an fppf covering of a scheme is effective. This follows immediately from the results discussed in this post, see Lemma Tag 04SK. You can find a detailed discussion in the chapter Examples of Stacks of the stacks project (in the stacks project we have only formulated this exact statement for the full subcategory of pairs X/U whose structure morphism X —> U is of finite type; this is due to our insistence to be honest about set theoretical issues).

Note how absurdly general this is! There are no assumptions on the morphisms X —> U at all. Now we can use this to show that suitable full subcategories of *Spaces* form stacks. For example, if we want to construct the stack parametrizing flat families of d-dimensional proper algebraic spaces, all we have to do is show that given an fppf covering {U_i —> U} of schemes and an algebraic space X —> U over U such that for each i the base change U_i \times_U X —> U_i is flat, proper with d-dimensional fibres, then also the morphism X —> U is flat, proper and has d-dimensional fibres. This is peanuts (compared to what goes into the theorem above).

Of course, to show that (under additional hypotheses on the families) we sometimes obtain an *algebraic* stack is quite a bit more work! For example you likely will have to add the hypothesis that X —> U is locally of finite presentation, which I intentionally omitted above, to make this work.

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