Every monoidal category $(\mathcal{C},\otimes)$ can be seen as a one-object bicategory: the morphisms are the objects of $\mathcal{C}$, and the $2$-morphisms are the morphisms of $\mathcal{C}$. In every bicategory, we can speak of left/right Kan extensions. Specifically, if $P$ is an object of $\mathcal{C}$ (the one "along which" we extend) and $F$ is another object of $\mathcal{C}$, then $\mathrm{Lan}\_P(F)$ is an object of $\mathcal{C}$ equipped with a morphism $\alpha : F \to \mathrm{Lan}\_P(F) \otimes P$, such that the evident universal property is satisfied: If $G$ is another object with a morphism $\beta : F \to G \otimes P$, then there is a unique morphism $\gamma :\mathrm{Lan}\_P(F) \to G$ with $(\gamma \otimes P) \circ \alpha = \beta$. The notation $\mathrm{Lan}\_P(F) = F/P$ makes sense, I would say: the left Kan extension tries to approximate this "quotient object".


Is this notion of a Kan extension in a monoidal category already known under a different name? Has it been studied, at least in some examples? I think that the *right* Kan extension is just the internal hom $[P,F]$. So by dualization, the left Kan extension is a kind of internal "co-hom".



Kan extensions inside a monoidal category
It is certainly the case that the duals of internal homs have appeared significantly less in the categorical literature. I've included a few more references below, but I am not sure this is a satisfying answer; I have not found any reference that lists many naturally-occurring examples.


In the cocartesian monoidal setting, these are known as *coexponential objects*, and are studied in papers such as Filinski's [Declarative Continuations and Categorical Duality](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.43.8729&rep=rep1&type=pdf), and Niefield–Wood's [Coexponentiability and Projectivity: Rigs, Rings, and Quantales](http://www.tac.mta.ca/tac/volumes/32/36/32-36abs.html).


More generally, monoidal categories with internal cohoms are usually called *coclosed*. Examples are studied in Wedhorn's [On Tannakian duality over valuation rings](https://www.sciencedirect.com/science/article/pii/S0021869304004107), Chikhladze–Lack–Street's  [Hopf monoidal comonads](http://web.science.mq.edu.au/%7Estreet/ChLaSt.pdf), and Silantyev's [Quantum Representation Theory and Manin matrices I: finite-dimensional case](https://arxiv.org/abs/2108.00269) (see §4.4.1 and Proposition 5.16 for instance), which refers to earlier work of Manin; and are made essential use of in Lyubinin's [Tannaka duality, coclosed categories and reconstruction for nonarchimedean bialgebras](https://arxiv.org/abs/1411.3183). However, in several of the explicit examples in the literature, coclosure follows from closure and duality, which makes for less interesting examples.


