The authors define the notions of \(C\)-universality for a class of spaces \(C\), and prove some results concerning the relation between strong universality for pairs and strong universality for spaces. In particular, they prove that if \(M\) is an ANR and \(X\) a homology dense subspace in \(M\) having SDAP, then for any pair \((K,C)\) the strong \((K,C)\)-universality of the pair \((M,X)\) implies the strong \(C\)-universality of \(X\). Conversely, if \(M\) is a Polish ANR and \(X\) a homotopy dense subset in \(M\), and if \(X\) is strongly universal for a \(2^\omega\)-stable weakly \(A_1\)-additive class \(C\), then \((M,X)\) is strongly \((M_0\cap C,C)\)-universal. The authors apply these results to prove enlarging, deleting and strong negligibility theorems for strongly universal and absorbing spaces.