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
: a product
is a universal arrow from
to
. The arrow comprises the projection maps.
More generally, given a small index category
, one may construct the functor category
, the objects of which are called diagrams. For each object
in
, there is a constant diagram
that maps every object in
to
and every morphism in
to
. The diagonal functor :{\mathcal {C}}\rightarrow {\mathcal {C}}^{\mathcal {J}}}
assigns to each object
of
the diagram
, and to each morphism
in
the natural transformation
in
(given for every object
of
by
). Thus, for example, in the case that
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
of
) is called a cone for
. These cones and their factorizations correspond precisely to the objects and morphisms of the comma category
, and a limit of
is a terminal object in
, i.e., a universal arrow
. Dually, a colimit of
is an initial object in the comma category
, i.e., a universal arrow
.
If every functor from
to
has a limit (which will be the case if
is complete), then the operation of taking limits is itself a functor from
to
. 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.