Take an arbitrary category $C$, and consider $End(C)$, any endofunctor category consisting of some or all endofunctors of $C$. I have several related questions:

- What restrictions must we impose on the base category $C$ such that $End(C)$ has a symmetric monoidal product?
- What restrictions must we impose on the functors $F \in Ob(End(C))$ for $End(C)$ to have a symmetric monoidal product?

I am also interested in the same two questions when we are interested in dagger compactness, ie, there is an endo functor $\dagger : End(C) \rightarrow End(C)$, where $\dagger$ is identity on objects but flips all the natural transformations in $End(C)$, ie all its arrows. $\dagger \circ \dagger = Id_{End(C)}$.

I am trying to build Abramsky, Coecke's semantic category for quantum protocols (also Selinger's construction) in terms endofunctor categories on arbitrary base categories.