It is the purpose of this note to show that existence results for vector-valued equilibria, of the type considered recently in [1] or [2],
can be deduced in a straightforward way from corresponding results about scalar-valued equilibria. We shall proceed as follows. First we prove an existence result for scalar-valued equilibria, employing a certain notion of generalized monotonicity, and from this we deduce several results for the vector-valued case, by using appropriate gauge functions.