Let $q : W \rightarrow V$ be a regular epimorphism in a category $\mathcal{C}$, and consider the pullback functor $q^* : \mathcal{C}/V \rightarrow \mathcal{C}/W$. Is $q^*$ necessarily essentially injective? That is, does $q^*C \cong q^*D$ imply $C \cong D$?
Context: I'm trying to get a better understanding of Schulman's stack semantics (https://arxiv.org/abs/1004.3802, definition 7.2). Given two objects $A,B$ in $\mathcal{C}/U$, we can consider the statement $U \Vdash (A \cong B)$; I wanted to know whether this is equivalent to just the statement that $A \cong B$. (Note: the formula $A \cong B$ is shorthand in categorical first order logic for $(\exists f : A \rightarrow B)(\exists g : A \rightarrow B)((g \circ f = \operatorname{Id}_B) \wedge (f \circ g = \operatorname{Id}_B))$.) If the answer to my question is "yes", then this equivalence becomes rather easy to show.
I know that it would be sufficient to show that $q^*$ is full and faithful. I suspect that the answer to my question is "no", because there doesn't seem to be any straightforward way to prove it. However, I don't have a good repertoire of examples that would allow me to construct a counter-example. If necessary, we can make additional assumptions on $\mathcal{C}$.