Given a category $\mathcal{C}$, it follows from Proposition 4.1 of Day's
Brian Day, Limit spaces and closed span categories, Lecture Notes in Mathematics, 420, 1974 (doi:10.1007/BFb0063100).
That the bicategory $\mathsf{Span}_\mathcal{C}$ of spans in $\mathcal{C}$ has right Kan extensions and right Kan lifts iff $\mathcal{C}$ is locally Cartesian closed.
- Does $\mathsf{Span}_\mathcal{C}$ also have left Kan extensions and left Kan lifts if (and perhaps only if?) $\mathcal{C}$ is locally Cartesian closed too?
- What would be explicit descriptions for $\mathsf{Ran}$ and $\mathsf{Rift}$ in $\mathsf{Span}_{\mathcal{C}}$? What about for $\mathsf{Lan}$ and $\mathsf{Lift}$ if these also exist?