Gabriel–Ulmer duality says that there is an equivalence of 2-categories (or in other words, a biequivalence)
where Lex is the 2-category of:
and LFP is the 2-category of
finitary right adjoint functors and
The idea is that an object $C \in Lex$ can be thought of as an essentially algebraic theory, which has a category of models $Lex(C,Set)$. Gabriel–Ulmer duality says that this category of models is locally finitely presentable, all LFP categories arise in this way, and we can recover the theory $C$ from its category of models.
The original source is:
A careful discussion and proof of the biequivalence is in
Some other general treatments of Gabriel-Ulmer duality (and generalizations to other doctrines):
C. Centazzo, E. M. Vitale, A duality relative to a limit doctrine, Theory and Appl. of Categories 10, No. 20, 2002, 486–497, pdf
Stephen Lack, John Power, Gabriel–Ulmer duality and Lawvere Theories enriched over a general base, pdf
M. Makkai, A. Pitts, Some results on locally finitely presentable categories, Trans. Amer. Math. Soc. 299 (1987), 473-496, MR88a:03162, doi, pdf
For a 2-dimensional analogue see the slides from a 2010 talk by Makkai: pdf
The version for $V$-enriched categories, where $V$ is closed symmetric monoidal category whose underlying category $V_0$ is locally small, complete and cocomplete is in section 9 (cf. theorem 9.8)
For a connection to Tannaka duality theory see
ncafe discussion here
Brian Day, Enriched Tannaka duality, JPAA 108 (1996) pp.17-22, MR97d:18008 doi