This talk will present hyperdoctrines, a widget invented by Lawvere in the late 70’s to give a categorical account of type theories. It has the advantage to dissociate every construction/rules of a type theory: structural rules, logical rules, quantifier rules, equality rules, etc. As a welcomed side effect, it questions the legitimacy of such rules. In particular, we will take some time to study the equality and the relevance of its usual definition, and try to give a feeling of Lawvere’s seminal thoughts on HoTT. If time permits, we shall discuss the insight of such an approach about a (for now non well established) “directed type theory”, by which is roughly meant a type theory in which paths between terms are not necessarily reversible.