Definition 7.3.0.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor between categories and let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory. We say that $F$ is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ if, for every object $C \in \operatorname{\mathcal{C}}$, the collection of morphisms $\{ F(u): F(C_0) \rightarrow F(C) \} _{u: C_0 \rightarrow C}$ exhibits $F(C)$ as a colimit of the diagram
\[ ( \operatorname{\mathcal{C}}^{0} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} ) \rightarrow \operatorname{\mathcal{C}}^{0} \hookrightarrow \operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}. \]