# Is there an official terminology for functors that are adjoint up to a given functor?

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.

• 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? – Kevin Carlson May 9 at 17:38
• Note that your condition is still related to the other : it is equivalent to $LU$ being a left adjoint to $R$ relatively to $U$. – Arnaud D. May 10 at 6:00
• @KevinCarlson: that's more or less exactly what I am observing. I added the example I have in mind to justify my question. – Ender Wiggins May 10 at 7:42
• @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 – Ender Wiggins May 10 at 7:44