  Motivated by team semantics and existential second-order logic, we develop a
model-theoretic framework for studying second-order objects such as sets and
relations. We introduce a notion of abstract elementary team categories that
generalizes the standard notion of abstract elementary class, and show that it
is an example of an accessible category. We apply our framework to show that
the logic $\mathsf{FOT}$ introduced by Kontinen and Yang satisfies a version of
Lindstr\"om's Theorem. Finally, we consider the problem of transferring
categoricity between different cardinalities for complete theories in
existential second-order logic (or independence logic) and prove both a
downwards and an upwards categoricity transfer result.
