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While the IPCC's physical science report usually assesses a handful of future scenarios, the IPCC Sixth Assessment Working Group III report (AR6 WGIII) on climate mitigation assesses hundreds to thousands of future emissions scenarios. A key task is to assess the global-mean temperature outcomes of these scenarios in a consistent manner, given the challenge that the emission scenarios from different integrated assessment models come with different sectoral and gas-to-gas coverage and cannot all be assessed consistently by complex Earth System Models. In this work, we describe the "climate assessment" workflow and its methods, including infilling of missing emissions and emissions harmonisation as applied to 1,202 mitigation scenarios in AR6 WGIII. We evaluate the global-mean temperature projections and effective radiative forcing characteristics (ERF) of climate emulators FaIRv1.6.2, MAGICCv7.5.3, and CICERO-SCM, discuss overshoot severity of the mitigation pathways using overshoot degree years, and look at an interpretation of compatibility with the Paris Agreement. We find that the lowest class of emission scenarios that limit global warming to "1.5°C (with a probability of greater than 50%) with no or limited overshoot" includes 90 scenarios for MAGICCv7.5.3, and 196 for FaIRv1.6.2. For the MAGICCv7.5.3 results, "limited overshoot" typically implies exceedance of median temperature projections of up to about 0.1°C for up to a few decades, before returning to below 1.5°C by or before the year 2100. For more than half of the scenarios of this category that comply with three criteria for being "Paris-compatible", including net-zero or net-negative greenhouse gas (GHG) emissions, are projected to see median temperatures decline by about 0.3-0.4°C after peaking at 1.5-1.6°C in 2035-2055. We compare the methods applied in AR6 with the methods used for SR1.5 and discuss the implications. This article also introduces a 'climate-assessment' Python package which allows for fully reproducing the IPCC AR6 WGIII temperature assessment. This work can be the start of a community tool for assessing the temperature outcomes related to emissions pathways, and potential further work extending the workflow from emissions to global climate by downscaling climate characteristics to a regional level and calculating impacts. [ABSTRACT FROM AUTHOR]