The merger of two white dwarfs (WDs) has for many years not been considered as the favoured model for the progenitor system of Type Ia supernovae (SNe Ia). But recent years have seen a change of opinion as a number of studies, both observational and theoretical, have concluded that they should contribute significantly to the observed SN Ia rate. In this paper, we study the ignition and propagation of detonation through post-merger remnants and we follow the resulting nucleosynthesis up to the point where a homologous expansion is reached. In our study we cover the entire range of WD masses and compositions. For the emergence of a detonation we study three different setups. The first two are guided by the merger remnants from our earlier simulations, while for the third one the ignitions were set by placing hotspots with properties determined by spatially resolved calculations taken from the literature. There are some caveats to our approach which we investigate. We carefully compare the nucleosynthetic yields of successful explosions with SN Ia observations. Only three of our models are consistent with all the imposed constraints and potentially lead to a standard Type Ia event. The first one, a 0.45 M⊙ helium (He) + 0.9 M⊙ carbon–oxygen (CO) WD system produces a sub-luminous, SN 1991bg-like event while the other two, a 0.45 M⊙ He+1.1 M⊙ oxygen–neon WD system and a 1.05 + 1.05 M⊙ system with two CO WDs, are good candidates for common SNe Ia.