Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.
Non-Existence of All Left Kan Extensions in $\boldsymbol {\mathsf{Rel}}$. Not all relations in $\boldsymbol {\mathsf{Rel}}$ admit left Kan extensions.
Characterisation of Relations Admitting Left Kan Extensions Along Them. The following conditions are equivalent:
The left Kan extension
along $R$ exists.
The relation $R$ admits a left adjoint in $\boldsymbol {\mathsf{Rel}}$.
The relation $R$ is of the form $f^{-1}$ (as in 
) for some function $f$.
Given relations $S\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}X$ and $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$, is there a characterisation of when the left Kan extension
exists in terms of properties of $R$ and $S$?
This question also appears as [Emily, Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II].
As shown in Item 2 of Proposition 9.2.1.1.1, the left Kan extension
along a relation of the form $R=f^{-1}$ exists. Is there a explicit description of it, similarly to the explicit description of right Kan extensions given in Proposition 9.2.3.1.1?
This question also appears as [Emily, Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II].
