Coend Collisions and Mono Preservation in Day Convolution

Day convolution is a fundamental construction for inducing monoidal structures on functor categories, yet its interaction with basic categorical properties such as monomorphisms is subtle. In particular, Day convolution does not preserve monomorphisms in general, even in the category of set-valued presheaves. In this paper we give a precise structural explanation of this failure. We construct a small counterexample showing that mono preservation can break already in very simple monoidal settings, and we identify the unique mechanism responsible for this phenomenon. We prove that failure of mono preservation is completely characterized by the presence of coend collisions, i.e. nontrivial identifications forced by the coend quotient in Day convolution. Building on this characterization, we derive an explicit system of sufficient conditions ensuring mono preservation. These conditions are formulated in terms of flatness of the promonoidal weight and a reduction to representable copresheaves ensuring practical checkability. Together, our results clarify when and why Day convolution preserves monomorphisms, and provide a usable criterion applicable in concrete categorical settings.