definition. diagonal functor [leinster2016basic, sec. 6.1] [tt-003T]

✍️source

Given a small category \({\cal J}\) and a category \({\cal C}\), the diagonal functor \[\Delta _{{\cal J}}: {\cal C} \to [{\cal J}, {\cal C}]\] maps each object \(X \in {\cal C}\) to the constant functor \(\Delta _{{\cal J}}(X): {\cal J} \to {\cal C}\), which in turn maps each object in \({\cal J}\) to \(X\), and all arrows in \({\cal J}\) to \(\mathit {1}_X\).

When \({\cal J}\) is clear in the context, we may write \(\Delta _{{\cal J}}(X)\) as \(\Delta _X\).

Particularly [kostecki2011introduction, 3.1, example 6], when \({\cal J}\) is a discrete category of two objects, \(\Delta : {\cal C} \to {\cal C} \times {\cal C}, \Delta (X)=(X, X)\) and \(\Delta (f)=(f, f)\) for \(f: X \to X^{\prime }\)

\(\Delta _{{\cal J}}(X)\) is the same as \(X \mathbin {\bullet } !\), thus [nakahira2023diagrammatic, eq. 2.12] \[\Delta _{{\cal J}} = - \mathbin {\bullet } !\]