Definition:Diagonal Morphism

Definition
Let $\CC$ be a category.

Let $f : X \to Y$ be a morphism in $\CC$.

Suppose that the fiber product $X \times_Y X$ of $f$ with $f$ exists.

The diagonal morphism $\Delta_f : X \to X \times_Y X$ is the morphism induced by the universal property of limits by $\operatorname {id}_X$ and $\operatorname {id}_X$.