Assume that we are given categories $\mathcal{A}, \mathcal{B}, \mathcal{C}$ and functors $U:\mathcal{A}\to\mathcal{B}$, $L:\mathcal{B}\to\mathcal{C}, R:\mathcal{C}\to \mathcal{B}$ such that $$\mathsf{Hom}_{\mathcal{C}}(LU(x),y)\cong \mathsf{Hom}_{\mathcal{B}}(U(x),R(y))$$ naturally in $x\in \mathcal{A}$ and $y\in \mathcal{C}$.

Question: Is there an appropriate terminology to describe this situation?

I would have been tempted to call $(L,R)$ adjoint functors relatively to $U$, but it seems that this terminology already exists (see here) and it denotes something different.

[Add-on, 10/05/19] For a concrete example of the above situation, assume that $B$ is a bialgebra over a commutative ring $\Bbbk$ which is also Frobenius as a $\Bbbk$-algebra and such that the Frobenius homomorphism is an integral on $B$ (these have been called FH-algebras by Pareigis). Denote by $\mathcal{M}$ the category of $\Bbbk$-modules and by $(-)^{\mathsf{co}B}:\mathcal{M}^B\to \mathcal{M}$ the functor sending a $B$-comodule $N$ to $$N^{\mathsf{co}B}:=\left\{n\in N\mid \delta_N(n)=n\otimes 1\right\}.$$ Consider also the functors $U_B:\mathcal{M}_B^B\to\mathcal{M}^B$ forgetting the action and $(-)^u:\mathcal{M}\to\mathcal{M}^B$ endowing a $\Bbbk$-module with the trivial comodule structure given by extending along the unit $u$ (ie, $\delta_V(v)=v\otimes 1$ for all $v\in V$). What one can prove now is that $(-)^u$ is always left adjoint to $(-)^{\mathsf{co}B}$ and that, under the foregoing hypothesis, we have an additional natural bijection $$\mathsf{Hom}\left(U_B(M)^{\mathsf{co}B},V\right)\cong \mathsf{Hom}^B\left(U_B(M),V^u\right)$$ (natural in $M\in\mathcal{M}_B^B$ and $V\in\mathcal{M}$).

This fact is connected with the study of the relationship between the Frobenius and the Hopf property for bialgebras, as well as the integral theory of the latter ones.

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    $\begingroup$ I'd say this notion is not so natural in that it has nothing to do with $\mathcal A$. You're just saying that $R$ looks like a right adjoint to $L$ on the essential image of $U$. Did you have some examples in mind? $\endgroup$ – Kevin Carlson May 9 at 17:38
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    $\begingroup$ Note that your condition is still related to the other : it is equivalent to $LU$ being a left adjoint to $R$ relatively to $U$. $\endgroup$ – Arnaud D. May 10 at 6:00
  • $\begingroup$ @KevinCarlson: that's more or less exactly what I am observing. I added the example I have in mind to justify my question. $\endgroup$ – Ender Wiggins May 10 at 7:42
  • $\begingroup$ @ArnaudD. You are right, of course. Since in the framework I am working I need/like to focus on $L$ and $R$, I didn't notice what you pointed out. Thanks $\endgroup$ – Ender Wiggins May 10 at 7:44

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