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Exercise 7.3.0.3 (Existence of Kan Extensions). Let $\operatorname{\mathcal{C}}$ be a category, let $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $F_0: \operatorname{\mathcal{C}}^{0} \rightarrow \operatorname{\mathcal{D}}$ be a functor between categories. Show that the following conditions are equivalent;

$(1)$

There exists a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ which is left Kan extended from $\operatorname{\mathcal{C}}^{0}$ and satisfies $F|_{ \operatorname{\mathcal{C}}^{0} } = F_0$.

$(2)$

For every object $C \in \operatorname{\mathcal{C}}$, the diagram

7.12
$$\label{equation:Kan-extension-existence} ( \operatorname{\mathcal{C}}^{0} \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/C} ) \rightarrow \operatorname{\mathcal{C}}^{0} \xrightarrow { F_0 } \operatorname{\mathcal{D}}$$

has a colimit in $\operatorname{\mathcal{D}}$.

Stated more informally, if the diagram (7.12) has a colimit in $\operatorname{\mathcal{D}}$, then that colimit depends functorially on the object $C \in \operatorname{\mathcal{C}}$.