On the detail: the distinction worth making early is between a tolerability problem, which is unpleasant and self-limiting, and a clinical problem, which is neither. They present differently and the thresholds for each are worth writing down before you need them.
Early satiety is not a side effect; it is the mechanism being observable. The useful distinction is between satiety, where you stop eating without distress, and aversion, where the thought of food is unpleasant. The first is the intended effect. The second frequently precedes the dose being too high or escalated too fast.
Injection-site reactions are more often diluent-related than peptide-related. Benzyl alcohol sensitivity is uncommon but real, and it presents as a consistent local reaction at every site with a preserved diluent and no reaction with an unpreserved one — which is a straightforward thing to establish. Injection depth is the other common cause: intradermal placement stings and welts, subcutaneous placement usually does not.
SURMOUNT-1 reported gastrointestinal events as the most frequent adverse events, mostly mild to moderate and mostly during escalation, with discontinuation for adverse events in the single digits per cent[1].
One qualification — reported incidence in a monitored trial population is a lower bound on what happens in an unmonitored one, because trial participants were escalated to protocol and supported through it.
If the timing does not fit the escalation, look for another explanation before settling on the drug.