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.

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 $$y \otimes z \circ \{ f, g \} \circ x = \{ y \circ f \circ \ell(x), z \circ g \circ r(x) \}$$ $$\{ h_\ell, h_r \} = h$$ $$\{ f, g \}_\ell = f$$ $$\{ f, g \}_r = g$$

rightexact, not left exact. Are you sure you want to be asking for a left adjoint here and not a right adjoint? $\endgroup$ – Noah Snyder Jan 3 at 20:28