Is there a name for monoidal categories $(\mathscr V, \otimes, I)$ such that $\otimes$ has a left adjoint $(\ell, r) : \mathscr V \to \mathscr V^2$? Have they been studied anywhere? What are some interesting examples?


A couple of remarks: when $I : 1 \to \mathscr V$ has a left adjoint, then $\mathscr V$ is semicartesian, i.e. the unit is terminal. When $\otimes$ has a left adjoint, which is furthermore the diagonal $\Delta : \mathscr V \to \mathscr V^2$, then $\mathscr V$ has binary products.




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I'll unwrap the definition here to make the structure more explicit. Let $(\mathscr V, \otimes, I)$ be a monoidal category. $\otimes$ has a left adjoint if we have the following.


* endofunctors $\ell : \mathscr V \to \mathscr V$ and $r : \mathscr V \to \mathscr V$;
* for every pair of morphisms $f : \ell(X) \to Y$ and $g : r(X) \to Z$, a morphism $\{f, g\} : X \to Y \otimes Z$;
* for every morphism $h : X \to Y \otimes Z$, morphisms $h\_\ell : \ell(X) \to Y$ and $h\_r : r(X) \to Z$,


such that, for all $x : X' \to X$, $y : Y \to Y'$ and $z : Z \to Z'$, we have
$$y \otimes z \circ \{ f, g \} \circ x = \{ y \circ f \circ \ell(x), z \circ g \circ r(x) \}$$
$$\{ h\_\ell, h\_r \} = h$$
$$\{ f, g \}\_\ell = f$$
$$\{ f, g \}\_r = g$$




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Edit (2021-09-27): these are *map pseudomonoids in $\mathrm{Cat}^{\mathrm{co}}$* in the terminology of Day–McCrudden–Street's [Dualizations and Antipodes](https://link.springer.com/article/10.1023%2FA%3A1024236601870). In [Kan extensions and cartesian monoidal
categories](https://arxiv.org/abs/1409.6405), Street defines a monoidal $\mathcal V$-category to be *cartesian* when it satisfies this condition. This terminology is justified at least when $\mathcal V = \mathrm{Set}$ by Tim Campion's answer. Preferably one would show it is correct whenever $\mathcal V$ has finite products.



Monoidal categories whose tensor has a left adjoint
Just to clean up the $\epsilon$ of room left after Qiaochu's answer -- we can get rid of the extra hypotheses. I'll write $I$ for the monoidal unit and $1$ for the terminal object.


Assume that $(\ell,r) \dashv \otimes$. Then the natural isomorphisms $A \cong I \otimes A \cong A \otimes I$ give rise, by adjunction, to maps $\ell A \to I$ and $r A \to I$, natural in $A$. We also have a unit map $A \to (\ell A) \otimes (r A)$, natural in $A$. Tensoring and composing, we get a map $A \to (\ell A) \otimes (r A) \to I \otimes I \cong I$, natural in $A$. That is, we have a cocone (with vertex $I$) on the identity functor for $V$. It follows that in the idempotent completion $\tilde V$ of $V$, there is a terminal object (which must be a retract of $I$).


Now, the idempotent completion $\tilde V$ again has a monoidal structure $\tilde \otimes$ with a left adjoint $(\tilde \ell, \tilde r)$. So the first part of Qiaochu's Eckmann-Hilton argument can be run in $\tilde V$: $I = I \otimes I = (I \times 1) \otimes (1 \times I) = (I \otimes 1) \times (1 \otimes I) = 1 \times 1 = 1$ (in the third expression, the products exist trivially, and in the fourth the product exists because $\otimes$ preserves products). That is, we must have $I\_{\tilde V} = 1\_{\tilde V}$. But $I\_{\tilde V}$ is the image of $I\_V$ in $\tilde V$, and the inclusion into the idempotent completion reflects terminal objects. Therefore $V$ has a terminal object, and $1\_V = I\_V$.


Then, as observed in the comments above, the second part of Qiaochu's Eckmann-Hilton argument can be run in $V$: $A \otimes B = (A \times 1) \otimes (1 \times B) = (A \otimes 1) \times (1 \otimes B) = A \times B$ (in the second expression, the products exist trivially, and in the third the product exists because $\otimes$ preserves products). That is, binary products exist in $V$ and agree with $\otimes$. In fact, the identity functor is an oplax monoidal functor from $(V,\otimes)$ to $(V,\times)$, which the argument shows is actually strong monoidal. Thus $(V,\otimes) \simeq (V,\times)$ as monoidal categories.


