I'm only just beginning to learn category theory, so apologies if this question has an obvious answer.
If I understand correctly, one of the reasons we care about universal properties is that they provide "nice" characterizations of objects whose specific, concrete construction might be rather unwieldy (free objects, tensor products, etc). But, showing that an object satisfying a given universal property actually exists in the first place typically requires recourse to these same unwieldy constructions.
Of course, this isn't too surprising, since a category doesn't contain any "new information" that wasn't already present in the concrete objects under consideration. But I'm wondering if there are any theorems that guarantee the existence of an object satisfying a given universal property, based on "simpler" purely category-theoretic criteria alone. By "simpler," I mean properties of a given category that might be easier or more natural to verify concretely than the messy construction of an object with a given universal property. It seems like such a result would also be necessary for showing the existence of objects satisfying a universal property in categories which are themselves constructed in purely category-theoretic terms.