nLab
direct limit

A direct limit is the same thing as a colimit. (Similarly, an inverse limit is the same thing as a limit.) In this context, a direct system is the same thing as a diagram, and a direct cone is the same thing as a cocone.

Many authors restrict this terminology to limits over directed sets (or filtered categories), especially the directed set $(\mathbb{N},\le )$ of natural numbers; see directed colimit (or filtered colimit) for discussion of this case if you think that it may be what you want.