Establishing an equivalence involves demonstrating strong similarities between the mathematical structures concerned.
In some cases, these structures may appear to be unrelated at a superficial or intuitive level, making the notion fairly powerful: it creates the opportunity to "translate" theorems between different kinds of mathematical structures, knowing that the essential meaning of those theorems is preserved under the translation.
However, in contrast to the situation common for isomorphisms in an algebraic setting, the composite of the functor and its "inverse" is not necessarily the identity mapping.
Due to this circumstance, a functor with these properties is sometimes called a weak equivalence of categories.
, or likewise, G is the right adjoint of F. Then C and D are equivalent (as defined above in that there are natural isomorphisms from FG to ID and IC to GF) if and only if
are not both full and faithful, then we may view their adjointness relation as expressing a "weaker form of equivalence" of categories.
Assuming that the natural transformations for the adjunctions are given, all of these formulations allow for an explicit construction of the necessary data, and no choice principles are needed.
The key property that one has to prove here is that the counit of an adjunction is an isomorphism if and only if the right adjoint is a full and faithful functor.
As a rule of thumb, an equivalence of categories preserves all "categorical" concepts and properties.
This group captures the essential "symmetries" of C. (One caveat: if C is not a small category, then the auto-equivalences of C may form a proper class rather than a set.)