Let $f : X \overset{f}{\rightarrow} Y \overset{g}{\rightarrow} \mathrm{Spec} (k)$ be morphisms of schemes (feel free to add any hypothesis necessary). Let $\mathrm{QCoh}(Y)$ denote the derived (stable) $\infty$-category of quasi-coherent sheaves, which carries a natural symmetric monoidal structure $\mathrm{QCoh}(Y)^\otimes \rightarrow \mathrm{Fin}_*$ whose tensor product is the derived tensor product of quasi-coherent sheaves. Then one can extend $f^* : \mathrm{QCoh}(Y) \rightarrow \mathrm{QCoh}(X)$ to a symmetric monoidal functor with a lax monoidal right adjoint $f_*$ and, of course, we have a similar situation for $g$.

Now, my question is twofold:

How do I define the $\mathrm{QCoh}(k)$-linear structure of $\mathrm{QCoh}(Y)$? I can think of two ways one could approach this: Firstly, one could exhibit $\mathrm{QCoh}(Y)^{\otimes, L} \rightarrow \mathrm{QCoh}(Y)^\otimes$ as tensored over itself (see 2.1.3 of DAGII) and them pull this map back along $g^* : \mathrm{QCoh}(k)^\otimes \rightarrow \mathrm{QCoh}(Y)^\otimes$ to obtain a map $M \rightarrow \mathrm{QCoh}(k)^\otimes$. Secondly, one could perhaps use a symmetric version $\mathbf{Mod}^\otimes$ of Lurie's $\infty$-operad $\mathbf{LM}^\otimes$, but here I fail to construct a corresponding coCartesian fibration $M' \rightarrow \mathbf{Mod}^\otimes$ of $\infty$-operads. Perhaps we do not even need an $\infty$-operad, and something like a coCartesian fibration $M'' \rightarrow \Delta^1 \times \mathrm{Fin}_*$ would do the trick?

Whatever approach chosen above should be robust enough to first realize $f^*$ as $\mathrm{QCoh}(k)$-linear and subsequently its adjoints $f_*$ and $f^!$ (where now additional hypotheses on $f$ are required, I'm sure), by applying some criterion for relative adjoints as developed in Higher Algebra, Section 7.3.2

I hope that someone can clear up my confusion. Thank you for your time!