Given a category $\mathcal X$ (that will be used as an underlying category) and collection of categories $\mathcal C(i)$ for $i \in I$ with a faithful functor from each $\mathcal C(i)$ to $\mathcal X$, a category $\mathcal C$ is called a fibered product of the diagram described if there is a faithful functor from it to every $\mathcal C(i)$ in a way that makes the diagram commute and it is a terminal object in the quasi-category of such sources / cones.

${\sf CAT}(\mathcal X)$ denotes the slice category of (the subcategory of ${\sf CAT}$ which contains only faithful functors) over $\mathcal X$.

- I guess that a product in ${\sf CAT}(\mathcal X)$ is a fibered product in ${\sf CAT}$, right?
- If so (*1), is ${\sf CAT}(\mathcal X)$ closed under products? / Is ${\sf CAT}$ closed under fibered product?
- In the known cases of merged / combined categories such as: ${\sf TopGrp}$, ${\sf TopVec}$, (and even ${\sf Rng}$, ${\sf Grp}$ themselves are actually non-trivial combinations of categories), is the fibered product equal / isomorphic / equivalent / higher order something to the merge of its components?