Following example 3.5.3, if \(\mathcal Z\) is a discrete (i.e. ordinary) commutative monoid \(Z\) and \(\mathbb{K}= Hk\) the Eilenberg-MacLane spectrum of an ordinary commutative ring \(k\), the \(\infty\)-category \(\mathrm{Mod}_{\mathbb{K}}^{\mathcal Z}\) is the unbounded derived \(\infty\)-category \(\mathcal D(\mathrm{mod}_k^Z)\) of the ordinary abelian \(1\)-category \(\mathrm{mod}_k^{Z}\coloneqq \mathrm{Fun}(Z, \mathrm{mod}_k)\) of \(Z\)-graded \(k\)-modules. This will be discussed in more detail in subsection 3.6.
Unpacking Day convolution from corollary 3.5.10 in these terms, the tensor product of an ordinary \(k\)-module \(M\) concentrated in degree \(z\in Z\) and an ordinary \(k\)-module \(N\) concentrated in degree \(w\in Z\) is given by the derived tensor product \(M\otimes_k^L N\) concentrated in degree \(z+w\in Z\).