I don't know any special terminology for regular lfp categories. The category of pointed sets is maybe the easiest example of the kind of coslice ca...

One suggestion:A presheaf topos is locally finitely presentable (you can find a proof here). The coslice category is then again locally finitely prese...

About the reverse implication (coincidentally, I looked at this last week): Suppose that $G$ is a discrete group acting on a topological space $X$, ...

Before you asked this question, my intuition was that essential subtoposes of a presheaf topos $\mathbf{PSh}(\mathcal{C})$ were all of the form $\math...