In category theory, a branch of mathematics, the diagonal functor is given by, which maps objects as well as morphisms. This functor can be employed to give a succinct alternate description of the product of objects within the category \mathcal{C}: a product a \times b is a universal arrow from \Delta to. The arrow comprises the projection maps. More generally, given a small index category \mathcal{J}, one may construct the functor category, the objects of which are called diagrams. For each object a in \mathcal{C}, there is a constant diagram that maps every object in \mathcal{J} to a and every morphism in \mathcal{J} to 1_a. The diagonal functor assigns to each object a of \mathcal{C} the diagram \Delta_a, and to each morphism in \mathcal{C} the natural transformation \eta in (given for every object j of \mathcal{J} by \eta_j = f). Thus, for example, in the case that \mathcal{J} is a discrete category with two objects, the diagonal functor is recovered. Diagonal functors provide a way to define limits and colimits of diagrams. Given a diagram, a natural transformation (for some object a of \mathcal{C}) is called a cone for \mathcal{F}. These cones and their factorizations correspond precisely to the objects and morphisms of the comma category, and a limit of \mathcal{F} is a terminal object in , i.e., a universal arrow. Dually, a colimit of \mathcal{F} is an initial object in the comma category, i.e., a universal arrow. If every functor from \mathcal{J} to \mathcal{C} has a limit (which will be the case if \mathcal{C} is complete), then the operation of taking limits is itself a functor from to \mathcal{C}. The limit functor is the right-adjoint of the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor. For example, the diagonal functor described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor.