I would like to define a category as $\bf Cat \downarrow Set$ (which would be the slice category of $\bf Cat$ over the object $\bf Set$. However, since $\bf Set$ is not an object of $\bf Cat$, I cannot do that. However, using the slice definition still gives us some category where the objects are $\bf Set$-valued functors and arrows are functors in $\bf Cat$ along with some 2-isomorphism.


So I don't think this is the result of some slice but it seems so close ! Moreover I could also extend this definition to all the categories, but again, since their is no category of all categories, I don't think it is the result of any slicing.




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My question is : is there a standard notation for these kind of construction ? Am I completely lost with those size issues?


Thanks a lot :)



Slice category over Set
What you're describing is the (2-)[comma category](https://en.wikipedia.org/wiki/Comma_category) $(\mathbf{Cat} \hookrightarrow \mathbf{CAT}) \downarrow (\mathbf{Set} : \mathbf{1} \to \mathbf{CAT})$, where $\mathbf{Cat}$ is the (2-)category of small categories, $\mathbf{1}$ is the terminal category and $\mathbf{CAT}$ is the (2-)category of locally small categories. The objects of this comma category are small categories with functors into $\mathbf{Set}$ and morphisms are functors between small categories commuting with the functors into $\mathbf{Set}$. Comma categories are a generalisation of (co)slice categories that often allow you to describe structure which is "slice-like", but isn't quite.


Alternatively, if you're happy with the objects being locally small categories, you can simply take the (2-)slice category $\mathbf{CAT}/\mathbf{Set}$.


