Kan extensions inside a monoidal category 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".
 A: 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, and Niefield–Wood's Coexponentiability and Projectivity: Rigs, Rings, and Quantales.
More generally, monoidal categories with internal cohoms are usually called coclosed. Examples are studied in Wedhorn's On Tannakian duality over valuation rings, Chikhladze–Lack–Street's  Hopf monoidal comonads, and Silantyev's Quantum Representation Theory and Manin matrices I: finite-dimensional case (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. However, in several of the explicit examples in the literature, coclosure follows from closure and duality, which makes for less interesting examples.
A: The common phrase is coexponential object.
The internal logic of coclosed categories is cointuitionistic logic. You can broadly think of cointuitionistic logic in terms of pattern matching, nondeterminism and control flow/exceptions/continuations. It's a well known trick that classically $\text{Set}^\text{op} = \text{CABA}$ the category of complete atomic boolean algebras which perhaps explains the nondeterminism connection a bit.
However coexponentials are perhaps not broadly studied because the weight of identities in a category with distributivity or exponentials collapses things to nonconstructive logic the same way as adding law of excluded middle. Only if you give up certain identities can you have both. You can kind of think of this as like exceptions let you observe the order of evaluation from "within the interpreter".
To deal with these kind of issues probably the most common approach is to use a substructural logic and use more controllable residuals/coresiduals instead. For example, composition is certainly not a Cartesian product. You can kind of see this as similar to monadic side effects. Recasting the language of monoids in terms of composition instead of Cartesian product is the same mechanism of allowing side effects by weakening identities.
