The IPCC's scientific assessment of the timing of net-zero emissions and 2030 emission reduction targets consistent with limiting warming to 1.5°C or 2°C rests on large scenario databases. Updates to this assessment, such as between the IPCC's Special Report on Global Warming of 1.5°C (SR1.5) of warming and the Sixth Assessment Report (AR6), are the result of intertwined, sometimes opaque, factors. Here we isolate one factor: the Earth System Model emulators used to estimate the global warming implications of scenarios. We show that warming projections using AR6-calibrated emulators are consistent, to within around 0.1°C, with projections made by the emulators used in SR1.5. The consistency is due to two almost compensating changes: the increase in assessed historical warming between SR1.5 (based on AR5) and AR6, and a reduction in projected warming due to improved agreement between the emulators' response to emissions and the assessment to which it is calibrated.