Sufficient conditions are provided for quantifier elimination to hold in a theory. These conditions are obtained by abstracting away the details present in several specific proofs of quantifier elimination. Category theory was used in some parts in order to make the conditions as general as possible so that they can be applied in a wide range of situations. The conditions have two main purposes:

1. to ensure that satisfaction of existential formulas is reflected by an embedding; and
2. to guarantee the existence of a minimal model extending specific models.

The first goal is obtained by requiring that a theory is $\exists$-adequate and the second by imposing the existence of an adjunction.

Joint work with João Rasga.