Is there a known method to write any category $ C $ as being equivalent to a slice category bundle $ \bar{C}_{/c}\to\bar{C} $, where $ C\simeq \bar{C}_{/c} $? It seems one can try to find a category $\bar{C}$ and some $ c\in\bar{C}$ with $\int \bar{C}[-,c] \to \bar{C} $ giving the appropriate bundle, where $\int$ denotes the category of elements. Is this a reasonable approach? Where can I read more about this problem? And, is there some kind of free construction that produces a bundle with the property I am looking for?

The motivation for my question is to find out when the correspondence taking objects in $C$ to slice bundles over $C$ can be reversed, so that I can transport problems about functors between slice bundles down to problems about objects in the base category.

Is there a known method to write any category $ C $ as being equivalent to a slice category bundle $ \bar{C}_{/c}\to\bar{C} $, where $ C\simeq \bar{C}_{/c} $? It seems one can try to find a category $\bar{C}$ and some $ c\in\bar{C}$ with $\int \bar{C}[-,c] \to \bar{C} $ giving the appropriate bundle, where $\int$ denotes the category of elements. Is this a reasonable approach? Where can I read more about this problem? And, is there some kind of free construction that produces a bundle with the property I am looking for?

The motivation for my question is to find out when the correspondence taking objects in $C$ to slice bundles over $C$ can be reversed, so that I can transport problems about functors between slice bundles down to problems about objects in the base category.

Edit for clarity. I am looking for what characterizes slice categories without the context of their domain projection. What intrinsic properties distinguish slice categories? As Todd pointed out, they always have terminal objects. Their colimits are reflected by their projection functors, so I imagine we don't have to worry about many colimits existing since the unsliced category may not have many colimits to begin with.