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.