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The category $H$ can be described as the category of $E$-valued sheaves on $D$, or $F$-valued sheaves on $C$. You get a site by taking the category $C \times D$ and taking the topology generated by th …

Is there any references on the tensor product of (locally) presentable categories ? All I know about this is Lurie's book that deals with the $\infty$-categorical version, and a few references that d …

This is true. It is for example easy to see that the full subcategory of objects of this form is closed under all finite colimits*, dense and that these objects are all finitely presentable. I unfortu …

The category $\mathbf{Sup}$ of sup-lattices (posets admitting all supremum and supremum preserving map between them) is a well known example of a $*$-autonomous category: The internal Hom is simply t …

Probably not the best one can do, and what follows might be a bit 'overkill', but it answer the question about dependency on universe, and it is a nice argument. Also if you know how the proof of th …

In combinatorial and accessible weak model categories (also on ArXiv) I've studied Bousfield localization of weak and semi-model categories in both the combinatorial and accessible case. In particular …