Categorical limit ($\lim$)

A limit of a diagram $D:\mathcal{J}\to \mathcal{C}$ is a terminal cone over $D$: every other cone factors through it uniquely.

A finite limit is a categorical limit whose indexing category has finitely many objects and morphisms.

A product of objects $(A_i)$ is a universal object equipped with projections to every $A_i$.

The equalizer of parallel arrows $f,g:A\rightrightarrows B$ is a universal arrow $e:E\to A$ satisfying $fe=ge$.