This study has the purpose of researching the main characteristics of the good university professor of postgraduates for educational researchers in order to build a perspective of the quality of the training they teach. The main difference with similar researches is that, in this case, the study used a pool of graduate school teachers from two of the most important universities of Latin America and Spain. The methodology for this research was based on a multiple case study strategy, with a questionnaire designed for this purpose, which was given to faculty members of graduate school at the UNAM and the UCM. Results show a high incidence in cognitive competence as a key feature to recognize a good teacher, as well as for how to become one of them. Faculty members also recommended focusing on social competence when trying to improve academic work. The conclusions question the fact that the postgraduate level is giving so much importance to Cognitive competence, when at a global level, it is being asked to rethink education considering humanistic criteria sensitive to social needs, more akin to Ethical competence and Social competence.