Is every abstract category a concrete category of structures? In conceptual mathematics by Lawvere and Schanuel, it says:

There are many more categories than just those given by abstract types of structure; however, those can be construed as full subcategories of the latter, so that the notion of map does not change.

Given a type of structure, one has structures of that type. Morphisms are structure preserving maps from one structure to another; composition is defined as in the category of sets.
Is every category isomorphic to a subcategory of a category of structures of a given type? Or how to interpret the above statement from the book?
Is there a precise notion of "type of structure" (the notion given in the book surely isn't the only one) such that every category is isomorphic to a concrete category of structures.


there are categories without a faithful functor to Set

There are in fact many such categories. https://arxiv.org/abs/1704.00303
