In "Relational Algebra by Way of Adjunctions," found at author's page (doi), section 2.4, an adjunction is described using the signature:
$$ L \dashv R:\mathscr{D}\to \mathscr{C}.$$
Based purely on how my understanding of type signatures work, the above states that the concept, $L \dashv R$, is an arrow from
$\mathscr{D}$ to $\mathscr{C}$–a Functor in this case. However, it seems to me that this may be a convenient (and seemingly standard) way to call out the relevant categories involved. From the adjunctions in the paper, $\eta$ and the component functors are utilized, but I don't see that $L \dashv R$ is ever actually used as a $\mathscr{D} \to \mathscr{C}$ functor in its own right.
On the other hand, I'm in no position to just assume that the authors didn't really mean what they wrote–these guys are good. So my question, is $ L\dashv R:\mathscr{D}\to \mathscr{C}$ a functor? What is the definition of that functor for $\Delta\dashv\times$, or any of the adjunctions in figure 3?
Best Answer
No, $L\dashv R$ is a potentially confusing shorthand for "$L: C\to D,R:D\to C$, and $L$ is left adjoint to $R$."