The bicategory $\mathsf{Rel}$ of sets, relations, and inclusions of relations has right Kan extensions and right Kan lifts¹, however I believe it does not have all left Kan extensions/lifts.

- Is it indeed the case that $\mathsf{Rel}$ does not have all left Kan extensions/lifts?
- (Moved to a sequel question)
- Is there an if and only if characterisation of the stronger condition of having $$\mathrm{Lan}_S\colon\mathsf{Rel}(A,X)\to\mathsf{Rel}(B,X)$$ exist for a relation $S\colon A⇸B$ in terms of properties of $S$?

¹I'll add a reference here with full proofs for this in the future.