Hello, I am used to a slightly different terminology, which seems appropriate.
A family of morphisms { x_i --> y }_{i \in I} in some category, all with the same codomain, is called a "sink" or a "cocone".
For a sink, as I know it, the codomain should also be specified, i.e. a sink is given by an object y and a family of morphisms x_i --> y. If I is not empty, this does not matter, but for empty I at least y should be given. A cocone is given by an object y and a natural transformation from some functor to the constant functor with value y; her y is also specified. So a sink is essentially a discrete cocone. A family {
x --> y_j }_{j \in J} all with the same domain is called a "source" or a "cone".
These are the dual notions.
Is there a name for a family of the form { x_i --> y_j }_{i \in I, j \in J} ? A cylinder? Or a frustrum (since I \neq J)?
I do not know. Where does it occur? Probably the domain and codomain should also be specified, possibly even an arrow. If a non-empty collection of arrows behave similarly (e.g. is mapped to the same arrow by a given functor F), this means the same a saying that they all behave in the same way as a given arrow 8e.g are mapped to some special morphism by F). A collection of two objects x,y (prescribed domain and codomain) is something different; it does not give an arrow Fx -->Fy. Greetings Reinhard [For admin and other information see: http://www.mta.ca/~cat-dist/ ]