I quote from Wikipedia, regarding the construction of the field of fractions of an integral domain:
"There is a categorical interpretation of this construction. Let $\mathcal{C}$ be the category of integral domains and injective ring maps. The functor from $\mathcal{C}$ to the category of fields which takes every integral domain to its fraction field and every homomorphism to the induced map on fields (which exists by the universal property) is the left adjoint of the forgetful functor from the category of fields to $\mathcal{C}$."
It is usually the case that forgetful functors have free functors as left adjoints. Hence I wonder, can we make sense of the phrase "the free field on an integral domain", and if so, is this structure the field of fractions?
I suggest you'd rather see this adjunction as a reflector-inclusion pair. $$Q:\mathbf{Dom_m} \dashv \mathbf{Field}:i$$
The forgetful functor is actually a full inclusion $i$ into the category of integral domains and monomorphisms, while the left adjoint $Q$ is the reflector "field of quotients". As is usual with full subcategories, you can think of the subcategory objects as having some extra property. So fields are integral domains with a special property and the $Q$ functor adds this special property to any integral domain. See also nlab