What relationship do two adjunctions imply? Suppose we have three categories $A,B,C$, and we have an adjunction between $A$ and $C$ and we also have another adjunction bewteen $B$ and $C$. What relationship does this imply between $A$ and $B$? Does this imply equivalence of the categories? Does it give a way to compute an adjunction between $A$ and $B$? There is a question about left and right. I am assuming that the composition of the functors in the adjunction between $A$ and $C$ generate a monad at $C$. Likewise, the composition of the functors in the adjunction between $B$ and $C$ generate a monad at $C$. I think this answers questions about left and right.

This doesn't imply very much at all. For instance, if $C$ is the terminal category, a left adjoint to the unique functor $A\to C$ is any functor $C\to A$ which sends the object of $C$ to an initial object of $A$. So to say that there are right adjoint functors $A\to C$ and $B\to C$ just means that $A$ and $B$ both have initial objects. That certainly doesn't imply $A$ and $B$ are equivalent, and doesn't even imply there exists any adjunction between $A$ and $B$.

One thing it does imply is that the nerves of $A$ and $B$ are homotopy equivalent, since the nerve of any adjoint functor is a homotopy equivalence. But this is a very weak condition (indeed, the nerve of any category with an initial or terminal object is contractible).