In category theory, a branch of mathematics, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}} is given by Δ ( a ) = ⟨ a , a ⟩ {\displaystyle \Delta (a)=\langle a,a\rangle } , 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 C {\displaystyle {\mathcal {C}}} : a product a × b {\displaystyle a\times b} is a universal arrow from Δ {\displaystyle \Delta } to ⟨ a , b ⟩ {\displaystyle \langle a,b\rangle } . The arrow comprises the projection maps.

More generally, given a small index category J {\displaystyle {\mathcal {J}}} , one may construct the functor category C J {\displaystyle {\mathcal {C}}^{\mathcal {J}}} , the objects of which are called diagrams. For each object a {\displaystyle a} in C {\displaystyle {\mathcal {C}}} , there is a constant diagram Δ a : J → C {\displaystyle \Delta _{a}:{\mathcal {J}}\to {\mathcal {C}}} that maps every object in J {\displaystyle {\mathcal {J}}} to a {\displaystyle a} and every morphism in J {\displaystyle {\mathcal {J}}} to 1 a {\displaystyle 1_{a}} . The diagonal functor Δ : C → C J {\displaystyle \Delta :{\mathcal {C}}\rightarrow {\mathcal {C}}^{\mathcal {J}}} assigns to each object a {\displaystyle a} of C {\displaystyle {\mathcal {C}}} the diagram Δ a {\displaystyle \Delta _{a}} , and to each morphism f : a → b {\displaystyle f:a\rightarrow b} in C {\displaystyle {\mathcal {C}}} the natural transformation η {\displaystyle \eta } in C J {\displaystyle {\mathcal {C}}^{\mathcal {J}}} (given for every object j {\displaystyle j} of J {\displaystyle {\mathcal {J}}} by η j = f {\displaystyle \eta _{j}=f} ). Thus, for example, in the case that J {\displaystyle {\mathcal {J}}} is a discrete category with two objects, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}} is recovered.

Diagonal functors provide a way to define limits and colimits of diagrams. Given a diagram F : J → C {\displaystyle {\mathcal {F}}:{\mathcal {J}}\rightarrow {\mathcal {C}}} , a natural transformation Δ a → F {\displaystyle \Delta _{a}\to {\mathcal {F}}} (for some object a {\displaystyle a} of C {\displaystyle {\mathcal {C}}} ) is called a cone for F {\displaystyle {\mathcal {F}}} . These cones and their factorizations correspond precisely to the objects and morphisms of the comma category ( Δ ↓ F ) {\displaystyle (\Delta \downarrow {\mathcal {F}})} , and a limit of F {\displaystyle {\mathcal {F}}} is a terminal object in ( Δ ↓ F ) {\displaystyle (\Delta \downarrow {\mathcal {F}})} , i.e., a universal arrow Δ → F {\displaystyle \Delta \rightarrow {\mathcal {F}}} . Dually, a colimit of F {\displaystyle {\mathcal {F}}} is an initial object in the comma category ( F ↓ Δ ) {\displaystyle ({\mathcal {F}}\downarrow \Delta )} , i.e., a universal arrow F → Δ {\displaystyle {\mathcal {F}}\rightarrow \Delta } .

If every functor from J {\displaystyle {\mathcal {J}}} to C {\displaystyle {\mathcal {C}}} has a limit (which will be the case if C {\displaystyle {\mathcal {C}}} is complete), then the operation of taking limits is itself a functor from C J {\displaystyle {\mathcal {C}}^{\mathcal {J}}} to C {\displaystyle {\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 C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}} described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor.