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Let $C$ be a presheaf category, and let $F : C^{m+1+n} \to C$ be a functor such that for any family of $m+n$ objects of $C$, $A_1,\dots,A_{m-i}$ and $B_1,\dots,B_{n+i}$ (letting $i$ vary between $0$ and $m$), $F(A_1,\dots,A_{m-i},-,B_1,\dots,B_{n+i})$ is a parametric left adjoint, that is, the induced functor
$L:C\to F(A_1,\dots,A_{m-i},\emptyset,B_1,\dots,B_{n+i})\downarrow C$admits a right adjoint.
Let $\Delta^{m+1+n}:C\to C^{m+1+n}$ denote the $m+1+n$-fold diagonal functor. Let $G=F\circ \Delta^{m+1+n}$. Then is it the case that the induced functor
$L_\Delta:C\to G(\emptyset)\downarrow C$
admits a right adjoint?
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