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 $[F,P]$. So by dualization, the left Kan extension is a kind of internal "co-hom".

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".

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 $\mathrm{Lan}_P(F) \otimes P \to F$, such that the evident universal property is satisfied: If $G$ is another object with a morphism $G \otimes P \to F$, then there is a unique morphism $\mathrm{Lan}_P(F) \to G$ such that the evident diagram commutes. 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 $[F,P]$. So by dualization, the left Kan extension is a kind of internal "co-hom".

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 $[F,P]$. So by dualization, the left Kan extension is a kind of internal "co-hom".