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Let $\mathrm{Cat}_\infty$ be the $\infty$-category of small $\infty$-categories, $\mathrm{Pr}^\mathrm{L}$ be the $\infty$-category of presentable $\infty$-categories, and $\mathcal P(\mathcal C)$ be the $\infty$-category of space-valued presheaves over any$\infty$-category $\mathcal C$. There is a functor $$\mathcal P:\mathrm{Cat}_\infty^{\mathrm{op}}\to \mathrm{Pr}^{\mathrm{L}}$$

sending $\mathcal C$ to $\mathcal P(\mathcal C)$ and a functor $f:\mathcal C\to \mathcal D$ to $$f^*:\mathcal P(\mathcal D)\to \mathcal P(\mathcal C),$$ the functor induced by precomposing with $f$.

Question 1: is it documented that this functor carries a canonical symmetric monoidal structure, with respect to the monoidal structure given by product of $\infty$-categories on the source* and the Lurie tensor product on the target?

Note also that for each $f:\mathcal C\to \mathcal D,f^*$ admits both a left and a right adjoint, given respectively by left and right Kan extension along $f$. This observation provides two functors $$\mathcal P_{(!)}: \mathrm{Cat}_\infty\to \mathrm{Pr}^{\mathrm{L}}$$

$$\mathcal P_{(*)}:\mathrm{Cat}_\infty\to \mathrm{Pr}^\mathrm{R}$$

Question 2: the same as question 1, but for $\mathcal P_{(!)}, \mathcal P_{(*)}$.

*This is not the Cartesian monoidal structure for $\mathrm{Cat}^\mathrm{op}_\infty$, I really mean the one whose operation is to take products. I guess it is the coCartesian monoidal structure for $\mathrm{Cat}^\mathrm{op}_\infty$.

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    $\begingroup$ I think that the answer to the version for $\mathcal P_{(!)}$ is covered in [HA, Rem 4.8.1.9]. $\endgroup$
    – Z. M
    Commented Apr 20, 2023 at 18:21
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    $\begingroup$ @Z.M It seems to me that HA 4.8.1.9 shows that the right adjoint of $P_{(!)}$ is lax symmetric monoidal. And HA 4.8.1.12 shows that the $P_{(!)}$ preserves commutative algebra objects. But I don't see an assertion that $P_{!)}$ is strong symmetric monoidal. $\endgroup$
    – Tim Campion
    Commented Apr 28, 2023 at 3:13
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    $\begingroup$ @TimCampion Typo, I wanted to refer to [HA, Rem 4.8.1.8]. $\endgroup$
    – Z. M
    Commented Apr 28, 2023 at 7:43
  • $\begingroup$ @Z.M Ah, great! I think that qualifies as an answer to the question! $\endgroup$
    – Tim Campion
    Commented Apr 28, 2023 at 11:11
  • $\begingroup$ Thanks for your comments! I guess 4.8.1.8 gives exactly symmetric monoidality for $P_{(!)}$, yes. And once we have this, can't we just argue that the equivalence Pr^L\simeq Pr^{R,op} is symmetric monoidal and deduce symmetric monoidality for the other two? $\endgroup$
    – W.Rether
    Commented Apr 28, 2023 at 15:05

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