The main result of the paper is the identification of a property that is bi-dual to exponential \(\kappa\)-domination. This property of a space~\(X\) states that if \(A\subseteq X\) and \(|A|\le2^\kappa\) then \(A\subseteq \bar B\) for some subset~\(B\) of~\(X\) of cardinality at most~\(\kappa\). The property bi-dual to this is \(\kappa\)-projectivity: if \(A\subseteq X\) and \(|A|\le2^\kappa\) then there is a continuous map \(f:X\to M\), where \(M\)~has weight at most~\(\kappa\) and \(f\)~is injective on~\(A\). The bi-duality of these properties means that \(X\)~has one iff \(C_p(X)\)~has the other, that is, \(X\)~has the domination property iff \(C_p(X)\)~has the projectivity property \emph{and} \(X\)~has the projectivity property iff \(C_p(X)\)~has the domination property. The author also exhibits various properties of \(\kappa\)-projective spaces, related to other cardinal functions and continuous maps. The paper concludes with a nice list of problems for the case \(\kappa=\aleph_0\).