nLab
finitely cocomplete category

A finitely cocomplete category is a category $C$ which admits all finite colimits, that is a colimit for any finite diagram $F:J\to C$. Finitely cocomplete categories are also called rex categories, although this is rarer than ‘lex’ for finitely complete categories. Compare a cocomplete category, which has all small colimits.

For further theory, see finitely complete category. Note that $C$ is finitely cocomplete if and only if its opposite category $C^{\mathrm{op}}$ is finitely complete.