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We give an explicit point-set construction of the Dennis trace map from the$K$-theory of endomorphisms $K\mathrm{End}(\mathcal{C})$ to topologicalHochschild homology $\mathrm{THH}(\mathcal{C})$ for any spectral Waldhausencategory $\mathcal{C}$. We describe the necessary technical foundations, mostnotably a well-behaved model for the spectral category of diagrams in$\mathcal{C}$ indexed by an ordinary category via the Moore end. This isapplied to define a version of Waldhausen's $S_{\bullet}$-construction forspectral Waldhausen categories, which is central to this account of the Dennistrace map.  Our goals are both convenience and transparency---we provide all detailsexcept for a proof of the additivity theorem for $\mathrm{THH}$, which is takenfor granted---and the exposition is concerned not with originality of ideas,but rather aims to provide a useful resource for learning about the Dennistrace and its underlying machinery.

Spectral Waldhausen categories, the $S_\bullet$-construction, and the Dennis trace