(cont) This has a right adjoint, the inverse image $f^*\colon P(Y) \to P(X)$ and $f^*$ has a right adjoint $f_!\colon P(X) \to P(Y)$, called the universal image and defined for a subset $A$ of $X$ by $f_!(A)$ consists of the elements of $Y$ all of whose inverse images lie in $A$.

Personal communication from Michael Barr: there is really no theory or history of double adjoints. Just that in some cases, a right (say) adjoint to a functor has a further right adjoint. The most obvious example of this is that a function $f\colon X \to Y$ induces the function direct image on the power sets (partially ordered sets considered as categories) $f_*\colon P(X) \to P(Y)$. (cont)