The embedding $I$ of $\ECat$ into $\REL{\ECat}$ admits a right adjoint.

Proof. Since for each $X,Y\in\Ob{\ECat}$ we have

we see that the functors $\REL{\ECat}\brk{I\prn{-},Y}$ are representable.

There is therefore an adjoint, whose value at $Y$ is $\Omega^Y$. Let us explain its value on morphisms: a morphism $\phi:X\to Y$ in $\ECat$ is sent to the morphism $\Omega^\phi:\Omega^X\to \Omega^Y$ in $\REL{\ECat}$ defined as the cartesian adjoint to the characteristic function of the subobject of $\Omega^X\times Y$ obtained by taking the image of the fibered product of $\in\Sub{X}$ and $R\Sub{\phi}$ over $X$.

In $\SET$, we find that $\Omega^\phi$ sends each $X\tick\subseteq X$ to $\brc{y \mid \prn{\exists x} \prn{x\in X\tick \text{ and } \phi\prn{x,y} \text{ is true}}}$.

If $\phi$ corresponds to the graph of an operation $f:X\to Y$, then $X\tick$ is sent to $\brc{y\mid \prn{\exists x}\prn{x\in X\tick \text{ and } y=f\prn{x}}}$. ∎