When considering the angular size of an object, the relevant distance to use is the emission distance, since the photon trajectories are set at that time and from then on are on radial paths towards us. This figure assumes the spatial geometry is flat, so we recover the Euclidean relation, using demit.
Because the emission distance goes through a minimum as a function of redshift, so the angular size of a fixed physical size also goes through a minimum.
Figure: Whittle's (upcoming) undergraduate textbook.