Sophie - sorry, \$$\mathcal{X}'\$$ should everywhere be \$$\mathcal{Y}\$$... or the other way around, but I think having two objects \$$x,x' \in \mathrm{Ob}(\mathcal{X})\$$ will be less confusing if there's no set \$$\mathcal{X}'\$$ lurking around, wishing that \$$x'\$$ belonged to it.