On endomorphisms of topological Hochschild homology

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