Let $\mathcal{C}$ be a category. A diagonal functor on $\mathcal{C}$ is a functor $\delta:\mathcal{C}\to \mathcal{C}^I$ for some set $I$ given by $$\delta(A)=(A)_{i\in I}\quad\mbox{ and }\quad \delta(\alpha)=(\alpha)_{i\in I}.$$ Here, $\mathcal{C}^I$ denotes the $I$ -fold direct product of the category $\mathcal{C}$ . For any given $I$ , $\delta$ is unique.

$\delta$ is faithful. Its image, $\delta(\mathcal{C})$ , is the subcategory of $\mathcal{C}^I$ whose objects are $(A)_{i\in I}$ and morphisms are $(\alpha)_{i\in I}$ . $\delta(\mathcal{C})$ is isomorphic to $\mathcal{C}$ , and may be pictured as the great diagonal of an $I$ -dimensional ``cube''.

More generally, when $I$ is a category, then the diagonal functor is just a functor $\delta$ that sends each object $A\in \mathcal{C}$ to the constant functor $\delta(A):I\to \mathcal{C}$ with fixed value $A$ , and every morphism $\alpha:A\to B$ to the natural transformation $\delta(\alpha):\delta(A)\dot{\to} \delta(B)$ , which sends every object $i\in I$ to $\alpha$ . A routine verification shows that $\delta(\alpha)$ is indeed a natural transformation.