Another questions about unidetermined monads.
EDIT: Here a note with a few more details on what they are: link
Let $T : C^o \to C$ be such a monad, so that the multiplication $\mu_A$ is determined by the unit from the equation $T\eta_A = \mu_A$.
Let moreover $T$ be lax idempotent, so that $T$ satisfies the following property:
let $a : TA\to A$ be an algebra; then $a\dashv \eta_A$ is an adjunction
(so that for example $a$ is uniquely determined up to iso).
Apply $T$ to this adjunction, and you will get an adjunction $Ta\dashv \mu_A$.
In the special case where $T$ is the presheaf construction $T$ not only reverses 1-cells, but also 2-cells; a similar definition applies.
For such a $T$, $\eta_A$ is the Yoneda embedding of $A$ and $T\eta$ the functor $[(TA)^o,Set]\to TA$ that restricts the functor $\lambda F.\zeta(F)$ to representables, giving a presheaf $\lambda a.\zeta(A(-,a))$.
But then its left adjoint $Ta$ must be determined by the universal property of $\text{Lan}_{\eta_A}$, and it turns out that if $F\in TA$ its yoneda extension acts as follows $$ \begin{align*} \text{Lan}_{\eta_A}F(G) &\cong \int^{a :A}Fa\times TA^o(\eta_A(a),G)\\ &\cong \int^{a:A}Fa\times TA(G,A(-,a))\\ &\cong \int^{a:A}Fa\times \mathcal O(G)(a) \end{align*} $$ where $\cal O$ is the Isbell functor.
In other words, $\text{Lan}_{\eta_A}F(-)\cong F\boxtimes \mathcal O(-)$, the functor tensor product of F, "twisted" by $\cal O$.
Even more suggestively, $\mu_A=T\eta_A$ has a right adjoint as well. It is the right extension along $\eta_A$, and a similar computation shows that
$\text{Ran}_\eta F$ is equal to the functor hom $\{\text{Spec}(F), G\}$ where again the Spec functor comes from Isbell duality.
I find the notation quite serendipitous[¹], I would be inclined to call the functor $G\mapsto F\boxtimes \mathcal O(G)$ the "extension of scalars" of ${\cal O}(G)$ by $F$, and the functor $G\mapsto \{\text{Spec}(F), G\}$ the functor that takes the "$F$-points of $G$".
What precisely am I after, here? Is it (part of) the reason why they're called Spec and $\cal O$?
[¹] A "serendipity" is the "fortuitous discovery of a pleasant truth, unexpectedly found while we were searching something else." I guess this definition applies here, where I was looking for something completely different, and I ended up suggesting a justification for the algebro-geometric notation for Isbell duality.