Let $\mathcal{C}$ be some category, $A \in \mathcal{C}$, $\mathbf{Grpd}_A$ be a groupoid consisting of all objects isomorphic to $A$ and all isomorphisms between them, and $i: \mathbf{Grpd}_A \to \mathcal{C}$ be the embedding. Are categories $\mathcal{C} \downarrow A$ and $\mathcal{C} \downarrow i$ necessarily equivalent?
I tried to construct the equivalence for a special case of bundles but there was an obstruction (one triangle that failed to be commutative), and I don't know how to approach the proof of inequivalence in general. Should I seek to prove inequivalence directly for some toy category (which I'd prefer not to do, see below) or is there an actual equivalence?
My motivation lies with bundles (so I'm mostly concerned with the special case $\mathcal{C} = \mathbf{Top}$): it seems 'evil' to fix one space as the base because we don't generally distinguish between isomorphic spaces, so a natural question arises: does it matter if we fix just one or the whole isomorphism class?
A hint would be nice, but if the question is actually too answer with only little knowledge of category theory (up to limits), a full answer is welcome too.