definition. evaluation [leinster2016basic, 6.2.4] [tt-0044]

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Let \({\cal S}\) be a small category, \({\cal C}\) a locally small category. For each \(X \in {\cal S}\), there is a functor \[ \begin {array}{lccc} \operatorname {ev}_X: & [{\cal S}, {\cal C}] & \to & {\cal C} \\ & \mathscr {F} & \mapsto & \mathscr {F}(X) \end {array} \] called evaluation at \(X\).

Given a diagram \(\mathscr {D}: {\cal J} \to [{\cal S}, {\cal C}]\), we have for each \(X \in {\cal S}\) a functor \[ \begin {array}{lccc} \mathscr {D} \mathbin {\bullet } \operatorname {ev}_X : & {\cal J} & \to & {\cal C} \\ & J & \mapsto & \mathscr {D}(J)(X) \end {array} \]

We write \(\mathscr {D} \mathbin {\bullet } \operatorname {ev}_X\) as \(\mathscr {D}(-)(X)\).