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Say, $B$ is a category fibered in groupoids over some category $C$, and $A$ is a category fibered in groupoids over $B$.

Suppose $A$ is a stack (over whatever site) over $B$, and $B$ is a stack over $C$, is $A$ also a stack over $C$?

"Conversely", if $A$ is a stack over $C$ and $B$ is a stack over $C$, then must $A$ be a stack over $B$?

Also, do properties like being algebraic, having a coarse moduli space, etc, depend on what you consider the base category to be? (Ie, if $A$ as a stack over $B$ is algebraic/has a CMS, then must $A$ as a stack over $C$ be algebraic/have a CMS as well?)

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OK, I think the first and second question are not sufficiently precise. I am going to assume you mean: C is a site and B is endowed with the topology inherited from C. In this case the Stacks project contains a lemma stating that the answer to your second question is "yes". See Lemma Tag 06NX.

Strangely, the statement that the composition A --> B --> C turns A into a fibred category over C if A --> B and B --> C define fibred categories (and similarly for categories fibred in groupoids and for stacks with topologies as above) seems to be missing from the Stacks project. I believe these are true, easier to prove than the lemma above, and they should be added there (now done).

An example of a statement of the kind you seem to be asking for in your final paragraph, in the setting of algebraic stacks, is Lemma Tag 05XY. Enjoy!

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