In 1925 Elie Cartan introduced the principal of triality
specifically for the Lie groups of type $D_4$, and in 1935 Ruth
Moufang initiated the study of Moufang loops. The observation of
the title in 1978 was made by Stephen Doro, who was in turn
motivated by the work of George Glauberman from 1968. Here the
author makes the statement precise in a categorical context. In
fact the most obvious categories of Moufang loops and groups with
triality are not equivalent, hence the need for the word
``essentially.''
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