. It is called the coslice category with respect to , or the category of objects under . The objects are pairs with . Given and , a morphism in the coslice category is a map making the following diagram commute: Arrow category and are identity functors on (so ). In this case, the comma category is the arrow category . Its objects are the morphisms of , and its morphisms are commuting squares in . Other variations In the case of the slice or coslice category, the identity functor may be replaced with some other functor; this yields a family of categories particularly useful in the study of adjoint functors. For example, if is the forgetful functor mapping an abelian group to its underlying set, and is some fixed set (regarded as a functor from 1), then the comma category has objects that are maps from to a set underlying a group. This relates to the left adjoint of , which is the functor that maps a set to the free abelian group having that set as its basis. In particular, the initial object of is the canonical injection , where is the free group generated by . An object of is called a morphism from to or a -structured arrow with domain . An object of is called a morphism from to or a -costructured arrow with codomain . Another special case occurs when both and are functors with domain . If and , then the comma category , written , is the discrete category whose objects are morphisms from to . An inserter category is a (non-full) subcategory of the comma category where and are required. The comma category can also be seen as the inserter of and , where and are the two projection functors out of the product category . Properties For each comma category there are forgetful functors from it. Domain functor, , which maps: objects: ; morphisms: ; Codomain functor, , which maps: objects: ; morphisms: . Arrow functor, , which maps: objects: ; morphisms: ; Examples of use Some notable categories Several interesting categories have a natural definition in terms of comma categories. The category of pointed sets is a comma category, with being (a functor selecting) any singleton set, and (the identity functor of) the category of sets. Each object of this category is a set, together with a function selecting