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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. Plain Language Summary: The IPCC's latest physical science report, the Working Group 1 Contribution to the Sixth Assessment Report (AR6), was released in August 2021. That report includes an update to the tools used to project the climate outcome of emission scenarios. Here we apply these newly calibrated tools, called earth system model emulators, to the set of scenarios assessed in the IPCC's Special Report on warming of 1.5°C (SR1.5). We find that two compensating changes lead to a remarkable consistency (peak warming projections within 0.1°C) between the projections made by the emulators used in SR1.5 and their descendants used in AR6. First, updates to the historical warming assessment since the SR1.5 (which was based on the IPCC's 2013 physical science report (AR5)) increase future warming projections. However, improved consistency between the emulators and the assessment of the underlying physics, particularly the short‐term warming response to emissions, lowers warming projections by an approximately equivalent amount. Our work reinforces the key messages from the IPCC: limiting warming to around 1.5°C is a great and urgent challenge, and it is up to us to decide whether we pull out all the stops to hold temperatures around 1.5°C or whether we sail on by. Key Points: Emulators used in IPCC Special Report on warming of 1.5°C and Sixth Assessment Report are remarkably consistent, despite their entirely new calibrationsThe consistency is due to two compensating factors: change in assessed historical warming and improvements to emulator calibration methods [ABSTRACT FROM AUTHOR]