This is in some sense a follow-up to previous questions I have had asking about the relationship between products and exponential objects.
Products can be written as, and in often are defined to be, the limits over diagrams from discrete categories.
In certain, but not most, cases, products and exponential objects may coincide. In contrast to products however, for which the definition seems easily formulable in terms of limits over a diagram, the definition of exponential objects seems to relate to a universal property, as does the definition of every limit, but does not seem formulable in terms of universal cones over diagrams.
I had the impression that universal properties/universal objects were strictly more general constructions in category theory than limits over a diagram, and if the exponential object really is a counterexample, than it would confirm my suspicion.
Are exponential objects examples of (co)universal objects which are not the (co)limits over any diagram? If they are not, are there any examples?