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Given a set $X$, is there a name (like $X$-concrete category) for those categories $\mathbf{C}$ equipped with a forgetful functor $F_x : \mathbf{C} \rightarrow \mathbf{Set}$ for each $x \in X$? The prototypical example being the category of models of a many-sorted theory whose sort symbols form a set $X$.

I am not specifically interested in the case where $\mathbf{Set}$ is the codomain; that was just an example.

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