Groupoid cores preserve pullbacks

Exercise 1.4.14. Show that the groupoid-core construction preserves pullbacks: for functors \(C\to E\leftarrow D\), construct a preferred equivalence \[ (C\times _E D)^{\simeq }\simeq C^{\simeq }\times _{E^{\simeq }}D^{\simeq }. \] Hint: Construct both directions from the groupoid-core and pullback universal properties, and compare the composites after projecting to the factors.