Remark 7.3.1.3 (02YA)

Remark 7.3.1.3. Stated more informally, a diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{\mathcal{C}}\ar [dr]^{F} \ar@ {=>}[]+<0pt,-25pt>;+<0pt,-50pt>^-{\alpha } & \\ K \ar [ur]^{\delta } \ar [rr]_{F_0} & & \operatorname{\mathcal{D}}. } \]

exhibits $F$ as a right Kan extension of $F_0$ along $\delta $ if, for every object $C \in \operatorname{\mathcal{C}}$, we can calculate the value $F(C) \in \operatorname{\mathcal{D}}$ as a limit of the diagram

\[ K_{C/} = \operatorname{\mathcal{C}}_{C/} \times _{\operatorname{\mathcal{C}}} K \rightarrow K \xrightarrow {F_0} \operatorname{\mathcal{D}}. \]

Note that this requirement characterizes the object $F(C) \in \operatorname{\mathcal{D}}$ up to isomorphism (see Proposition 7.1.1.12). We will later prove a stronger assertion: if the diagrams $\delta : K \rightarrow \operatorname{\mathcal{C}}$ and $F_0: K \rightarrow \operatorname{\mathcal{D}}$ are fixed, then a right Kan extension of $F_0$ along $\delta $ is uniquely determined (up to isomorphism) as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ (Remark 7.3.6.6).