Proposition 8.5.12.1.1 (02XF): Retractions in $\mathsf{Rel}$

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

The following conditions are equivalent:

1.
The relation $R$ is a retraction in $\mathsf{Rel}$.
2.
We have $R=\operatorname {\mathrm{Gr}}(f)$ for some injective function $f\colon A\to B$.

Proof of Proposition 8.5.12.1.1.

Omitted.

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