The definition of THH for Waldhausen categories

Let $\mathcal{C}$ be a Waldhausen category, and let $\operatorname{End}(\mathcal{C})$ be its category of endomorphisms. Let $\widetilde{K}(\operatorname{End}(\mathcal{C}))$ be the reduced algebraic $K$-theory spectrum, with Verschiebung filtration $V^{\geq i}$. The Waldhausen-category THH proposal. The definition of topological Hochschild homology of $\mathcal{C}$ should be

$$\operatorname{THH}(\mathcal{C}):=\widetilde{K}(\operatorname{End}(\mathcal{C}))/V^{\geq 2}.$$

This is proposed because general Waldhausen categories lack a canonical definition of topological Hochschild homology, whereas the Verschiebung construction on their endomorphism categories is available; the source presents it as a proposal rather than an established definition.