We are dealing with projective classes (in short PC) over first-order vocabularies with no restrictions on the (possibly infinite) arities of relation or operation symbols. We verify that PC(L_{\infty\lambda})=RPC(L_{\infty\lambda}) for any infinite cardinal \lambda, and that if \lambda, is singular, then PC(L_{\infty\lambda})=PC(L_{\infty\lambda^+}). If \lambda is regular, then a class of structures over a \lambda-ary vocabulary is PC(L_{\infty\lambda})-definable iff it is the image of a \lambda-continuous functor on a \lambda-accessible category. It follows that many PC classes of structures, previously known not to be closed under elementary equivalence over any L_{\infty\lambda}, are not co-PC over L_{\infty\infty}. Those classes arise from diverse contexts including convex l-subgroup lattices of lattice-ordered groups, ideal lattices of rings, nonstable K_0-theory of rings, coordinatization of sectionally complemented modular lattices, and real spectra of commutative unital rings. For example, the class of posets of finitely generated two-sided ideals of all unital rings is PC but not co-PC over L_{\infty\infty}. We also provide a negative solution to a problem, raised in~2011 by Gillibert and the author, asking whether essential surjectivity of a ``well-behaved'' functor on objects entails its essential surjectivity on diagrams indexed by arbitrary finite posets.