Proposition 7.2.2.1.1 (00NQ): Left Kan Lifts in $\textbf{Rel}$

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Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation.

Non-Existence of All Left Kan Lifts in $\textbf{Rel}$. Not all relations in $\textbf{Rel}$ admit left Kan lifts.
Characterisation of Relations Admitting Left Kan Lifts Along Them. The following conditions are equivalent:
1. The left Kan lift
  \[ \text{Lift}_{R}\colon \mathbf{Rel}\webleft (X,B\webright )\to \mathbf{Rel}\webleft (X,A\webright ) \]
  
  along $R$ exists.
2. The relation $R$ admits a right adjoint in $\textbf{Rel}$.
3. The relation $R$ is of the form $\text{Gr}\webleft (f\webright )$ (as in Definition 7.3.1.1.1) for some function $f$.