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The first sentence does not type-check:
Sets in homotopy type theory behave a lot like the internal language of a topos.
What you mean is something like:
The syntax for sets in homotopy type theory behaves a lot like the internal language of a topos.
I am taking the liberty of truncating the title from semantics of Set in homotopy type theory to Set in homotopy type theory.
Besides better matching the corresponding entry Set, and being less bulky as a title, this matches the title of the single reference that is given, and is not as directly in conflict with the idea offered in the Idea-section.
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