The multicategory of Waldhausen categories is "enriched over itself": the Hom-set of $k$-exact functors can be given a Waldhausen category structure by letting the morphisms be natural transformations, and cofibrations/weak equivalences be levelwise. With this construction, $k$-fold composition is a $k$-exact functor $$\mathrm{Hom}(\mathcal{E}_{k-1}, \mathcal{E}_k) \times\cdots\times \mathrm{Hom}(\mathcal{E}_0, \mathcal{E}_1) \longrightarrow \mathrm{Hom}(\mathcal{E}_0, \mathcal{E}_k).$$ (It also works for general composition, not just on the $1$-level, but I thought that writing that out here would not be worthwhile.)

This does not seem to be (a) new or (b) an example of a new construction, but I am having trouble finding references for it. Can anyone help?