A bi-pointed object in a category with terminal object is a co-span from to itself, i.e. a diagram
Similarly, a pointed object in a category with initial object and terminal object is a co-span from to , and if has in addition binary coproducts then a bi-pointed object in is the same as a co-span from to .
From the bicategory structure on co-spans in bi-pointed objects in naturally inherit the structure of a monoidal category
Assume that the terminal object is the tensor unit in .
Then moreover, following the construction of the -internal hom of pointed objects and being a special case of that of co-spans in , there is an internal hom-object of bipointed objects and defined as the pullback