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Let $\mathcal{E}$ be a category with finite limits, and let $Arr : Cat(\mathcal{E}) \to \mathcal{E}$ be the forgetful functor that sends an internal category to its object of arrows/morphisms. Does this functor always have a right adjoint? In particular, does $Arr : Cat \to Set$ have a right adjoint?

One conjecture I have is that if $\mathcal{E}$ also has binary coproducts and $X$ is an object of $\mathcal{E}$, then the right adjoint to $Arr$ would send $X$ to the indiscrete internal category on the $\mathcal{E}$-object $X + X$ of objects.

I am particularly interested in knowing whether $Arr : Cat(\mathcal{E}) \to \mathcal{E}$ preserves (binary) coproducts in general, which is true in the case $\mathcal{E} = Set$.

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    $\begingroup$ I suppose a related question would be: if $\mathcal{E}$ has binary coproducts, then does $Cat(\mathcal{E})$ have them as well, and do we know how they are constructed? $\endgroup$
    – User7819
    Jan 22, 2021 at 16:43

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No, $Arr$ does not even have a right adjoint in the case $\mathcal{E}=\mathsf{Set}$. The reason is that it fails to preserve coequalizers.

Any monoid $M$ can be viewed as a category with one object (so the monoid is the set of arrows under composition). The coequalizer of a pair of monoid homomorphisms in the category $\mathsf{Mon}$ of monoids agrees with the coequalizer of the corresponding pair of functors in $\mathsf{Cat}$. But the forgetful functor from $\mathsf{Mon}$ to $\mathsf{Set}$ does not preserve coequalizers, so neither does $Arr$.

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