In mathematics, specifically in homotopy theory and (higher) category theory, coherency is the standard that equalities or diagrams must satisfy when they hold "up to homotopy" or "up to isomorphism".
The adjectives such as "pseudo-" and "lax-" are used to refer to the fact equalities are weakened in coherent ways; e.g., pseudo-functor, pseudoalgebra.
But in some cases, such as prestacks, there can be several canonical isomorphisms and there might not be an obvious choice among them.
Replacing coherent isomorphisms by equalities is usually called strictification or rectification.
[3] Each of them has the rough form that "every weak structure of some sort is equivalent to a stricter one".