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Author of this page: I’m not entirely sure what a good name for such a model would be, I don’t really like the current name myself. I chose “concrete” in part because in the categorical semantics of the existence of collections of elements in the model category $C$ is basically the same as a faithful functor $U:C \to Set$, which is the definition of a concrete category, but mostly because I am terrible at coming up with a better name for this model. So if you have any other suggestions for names feel free to list them out.
To elaborate on the “concrete” part, the sets of morphisms of any category are already defined as a functor $Mor:C \times C \to Set$, function composition $\circ_{A,B,C}:Mor(B,C) \times Mor(A, B) \to Mor(A,C)$ is already defined as a family of functions in $Set$, and a concrete category is defined as a category with a functor $U:C \to Set$, so one could append to the concrete category the structure of function evaluation defined as a family of functions $\epsilon_{A,B}:Mor(A, B) \times U(A) \to U(B)$ in $Set$ satisfying certain axioms.
Clarifying the last sentence in my previous comment: many (but not all) concrete categories (Set being one of them, but also Ab, CRing, Top, Met, Diff, et cetera) have a function evaluation structure $\epsilon_{A,B}:Mor(A,B) \times U(A) \to U(B)$ as described in the last sentence of the previous comment, which comes from the fact that group homomorphisms/ring homomorphisms/continuous functions/isometries/differentiable functions are functions of sets which could evaluate an element of a domain to return an element in the codomain.
Of course, in other concrete categories (such as Rel), such a structure doesn’t exist, since relations aren’t functions.
I don’t understand post #5; the faithfulness of $U : C \to Set$ means that you lose nothing by reinterpreting morphisms $A \to B$ of $C$ as functions $U(A) \to U(B)$.
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