A (di)graph \( G \) is \({homomorphically; full}\) if every homomorphic image of \( G \) is a sub(di)graph of \( G \). This class of (di)graphs arose in the study of whether a homomorphism from a given graph \( G \) to a fixed graph \( H \) can be factored through a fixed graph \( Y \). Brewster and MacGillivray proved that the homomorphically full irreflexive graphs are precisely the graphs that contain neither \( P_4 \) nor \( 2K_2 \) as an induced subgraph. In this paper, we show that the homomorphically full reflexive graphs are precisely threshold graphs, i.e., the graphs that contain none of \( P_4 \), \( 2K_2 \), and \( C_4 \) as an induced subgraph. We also characterize the reflexive semicomplete digraphs that are homomorphically full, and discuss the relationship of these digraphs and Ferrers digraphs.