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Are there examples of faithful functors $F:\mathbf{Rel}\to \mathbf C$, where C is a concrete category over Set? Or can it be proved that such functors don't exists?

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$\mathrm{Rel}$ is itself concretizable by the functor $U : \mathrm{Rel} → \mathrm{Set}$ defined on objects by $UA = 2^A$, and on morphisms as a generalized image. Given a relation $R ⊆ A × B$ and a subset $X ⊆ A$, set $UR(X) = \{b ∈ B : \exists x ∈ X, \; xRb\}$.

To get other examples of faithful functors $\mathrm{Rel} → \mathscr C$, you can now compose $U$ with faithful functors $\mathrm{Set} → \mathscr C$.

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