We can define internal categories in a monoidal category like [this](http://ncatlab.org/nlab/show/internal+category+in+a+monoidal+category). Let $C$ be a dagger symmetric monoidal category. Will $C$ be locally finitely presentable? Let $C\_{int}$ be the category of internal categories in $C$. Is $C\_{int}$ locally finitely presentable?



lfp property for dagger symmetric monoidal categories and their internal categories
For a [dagger-category](http://ncatlab.org/nlab/show/dagger-category) $C$, we have $C^{op} \simeq C$. But the only time that a category $C$ and its opposite can both be locally finitely presentable is when $C$ is a poset (this is covered in the text by Adámek and Rosicky). So the answer to the first question is: only when $C$ is an algebraic lattice. 


The category of internal categories in an lfp category is again lfp. (Because the category of internal categories will be monadic over the category of internal graphs, and the latter is lfp.) However, a poset won't have any interesting internal categories. 


