On endomorphisms of topological Hochschild homology

We compute endomorphisms of topological Hochschild homology ($\mathrm{THH}$) as a functor on stable $\infty$-categories, as well as variants thereof: we also compute endomorphisms of the $k$-linear Hochschild homology functor $\mathrm{HH}_k$ over some base $k$; and endomorphisms of $\mathrm{THH}$ as a functor on stably symmetric monoidal $\infty$-categories.