Terminology: (Co)equalizer of a family of morphisms?

Question

Let $A,B\in \mathsf{C}$ be two objects in a category $\mathsf{C}$. Let $(f_i\colon A\rightarrow B)_{i\in I}$ be a family of parallel morphisms in $\mathsf{C}$. Consider the corresponding diagram in $\mathsf{C}$. If $I$ is a two-element set, one calls a (co)limit over this diagram a (co)equalizer of the two morphisms. Is there a name for a (co)limit of a general diagram of parallel morphisms indexed by an arbitray set $I$? Maybe it is called a "(co)equalizer of a family of morphisms"?

Answer 1

Yes, it can be called the (joint) coequaliser of the family of morphisms.

To prove I didn’t just make that up, see e.g. the language used in this definition. If your category admits coproducts of the necessary size, the joint coequaliser is a special case of the usual coequaliser via the known formula for colimits.