Given a functor$F: C\to D$ we say that $F$admits a proadjoint if the canonical extension $pro(F): pro(C)\to pro(D)$ of $F$ to the categories of pro-objects has a left adjoint$G$. In other words, there is a functor $G: pro(D)\to pro(C)$ and a bijection

$pro(C)(GY',X) \cong pro(D)(Y',y(F)X)$

natural in $X$ and $Y'$, where $y:C\hookrightarrow pro(C)$ is the Yoneda embedding into the category of proobjects $pro(C)\subset Set^{C^{op}}$. Equivalently, for every prorepresentable functor$X:C'^{op}\to Set$, the functor $X\mapsto X\circ F$ is also prorepresentable.

J.-M. Cordier, T. Porter, Shape theory : Categorical Methods of Approximation, (sec. 2.3), Mathematics and its Applications, Ellis Horwood Ltd., March 1989, 207 pages.Dover addition (2008) (Link to publishers here)