A category $C$ satisfying (any of) the following equivalent conditions is said to be locally finitely presentable (or lfp):

$C$ has all small colimits, the category $C_{fp}$ is essentially small, and any object in $C$ is a filtered colimit of the canonical diagram of finitely presentable objects mapping into it.

$C$ is the category of models for an essentially algebraic theory. Here an ‘essentially algebraic theory’ is a small category $D$ with finite limits, and its category of ‘models’ is the category of finite-limit-preserving functors $D \to Set$. (See Gabriel–Ulmer duality.)

$C$ is the category of models for a finite limit sketch.

$C_{fp}$ has finite colimits, and the restricted Yoneda embedding$C\hookrightarrow [C_{fp}^{op},Set]$ identifies $C$ with the category of finite-limit-preserving functors $C_{fp}^{op} \to Set$.