This paper introduces a class of oplax endofunctors on bicategories and studies their associated coherence structures, factorization behavior, and idempotence properties. We define a hierarchy of coherence conditions for oplax endofunctors and analyze how non-invertibility of structural 2-cells affects stability under composition and factorization. In particular, we show that failures of invertibility in coherence data lead to instability of factorization systems and obstruct idempotent collapse. We further distinguish between fully coherent pseudofunctors and non-invertibly oplax endofunctors, establishing that idempotence is equivalent to global coherence stability across all levels of the hierarchy. The results provide a structural classification of oplax endofunctors in terms of coherence preservation and iteration behavior.
Yugo Hidaka (Thu,) studied this question.