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:
small finitely complete categories,
finite limit$\:$ preserving functors, and
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. There are similar dualities for other classes of theory such as regular theories.
A version of Gabriel–Ulmer duality for enriched category theory was proved by Max Kelly (see LackTendas). Let $\mathcal{V}$ be a symmetric monoidal closed complete and cocomplete category which is locally finitely presentable as a closed category. Then let $\mathcal{V}$-$Lex$ be the 2-category of finitely complete $\mathcal{V}$-categories ($\mathcal{V}$-categories with finite weighted limits), finite limit preserving $\mathcal{V}$-functors, and $\mathcal{V}$-natural transformations, and $\mathcal{V}$-$LFP$ the 2-category of locally finitely presentable $\mathcal{V}$-categories, right adjoint $\mathcal{V}$-functors that preserve filtered colimits, and $\mathcal{V}$-natural transformations. Then there is a biequivalence
For instance, in the truth value-enriched case, the duality is between meet semilattices and algebraic lattices.
Gabriel-Ulmer duality is a duality exhibited by the 2-Chu construction, $Chu(Cat,Set)$.
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
A formal-categorical account using Yoneda structures can be found in
For a discussion of Gabriel–Ulmer duality and related dualities in the context of enriched category theory see
This discusses (see Theorem 2.1) Kelly’s original result for $V$-enriched categories, where $V$ is a closed symmetric monoidal category whose underlying category $V_0$ is locally small, complete and cocomplete, in section 9 (cf. theorem 9.8) of
For a connection to Tannaka duality theory see
nCafé discussion here
Brian Day, Enriched Tannaka duality, JPAA 108 (1996) pp.17-22, MR97d:18008 doi
For a discussion of an $\infty$-version of Gabriel-Ulmer duality between finitely complete and idempotent complete $(\infty, 1)$-categories and locally finitely presentable $(\infty, 1)$-categories see this MO discussion.