$\require{AMScd}$Kučera, JPAA 1971 shows the remarkable result that every category is a quotient of a concrete one:

Given a category $\cal K$ there is a category $\check{\mathcal{K}}$ which is concrete and endowed with a (super-highly non unique) congruence $\sim$ such that the category having the same objects of $\check{\mathcal{K}}$ and $\hom_{\check{\mathcal{K}}}(K_1, K_2)/\!\!\sim$ as sets of morphisms is equivalent (in fact isomorphic) to $\cal K$.

Given that a particular example of a congruence is the homotopy relation on the cofibrant-fibrant objects of a model category, it seems natural to wonder if there is a finer result that classifies which categories are *localization* of concrete ones.

If I'm not wrong it is fairly easy to show that the class $\mathcal{W}_\sim$ of arrows in $\check{\mathcal{K}}$ defined by $$ \Big\{ f\mid \exists g,g' : (fg\sim 1) \land (g'f\sim 1) \Big\} $$ is a 2-out-of-6 class. Then it seems natural to consider the localization $\check{\mathcal{K}}[\mathcal{W}_\sim^{-1}]$ and to consider the span $$ \begin{CD} \check{\mathcal{K}} @>>> \check{\mathcal{K}}[\mathcal{W}_\sim^{-1}] \\ @VVV \\ \mathcal K \end{CD} $$ which seems to be canonically[1] built out of $\mathcal K$.

[1] I said that the construction given in Kučera's paper is not canonical; but I think you can refine it to let the correspondence $\mathcal{K}\mapsto \check{\mathcal{K}}$ a functor (and also that this functor is part of an adjunction).