nLab
multilimit

A multilimit of a diagram $F:D\to C$ in a category $C$ is a (small) set $L$ of cones over $F$ such that any cone over $F$ factors uniquely through a unique element of $L$. Of course, limits are precisely multilimits for which $L$ is a singleton.

For example, the category of fields has no initial object, but it has a multi-initial-object, namely the family of all prime field?s.