\begin{definition}[Definition:Diagonal Morphism]
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 $\operatorname {id}_X$.
Category:Definitions/Category Theory
\end{definition}