Remark 7.3.6.8 (030H)

Remark 7.3.6.8. In the situation of Proposition 7.3.6.7, the horizontal maps in the diagram (7.28) are Kan fibrations (Corollary 4.1.4.2 and Proposition 4.6.1.21). Consequently, the diagram (7.28) is a homotopy pullback square if and only if the induced map

\[ \xymatrix@C =50pt{ \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G ) \ar [d]^{\theta } \\ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) }( F_0, G_0) \times _{ \operatorname{Hom}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) }( UF_0 , UG_0) } \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) }( UF, UG)} \]

is a homotopy equivalence (Example 3.4.1.3). Writing $\operatorname{\mathcal{M}}$ for the fiber product

\[ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}(\operatorname{\mathcal{C}}^{0}, \operatorname{\mathcal{E}}) } \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]

and $V: \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{M}}$ for the functor given by $V(H) = ( H|_{\operatorname{\mathcal{C}}^{0}}, U \circ H)$, we can identify $\theta $ with the map $\operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})}( F, G) \rightarrow \operatorname{Hom}_{ \operatorname{\mathcal{M}}}( V(F), V(G) )$ determined by $V$. We can therefore restate Proposition 7.3.6.7 as follows:

If the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $U$-left Kan extended from $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$, then it is $V$-initial when viewed as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.
If the functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $U$-right Kan extended from $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$, then it is $V$-final when viewed as an object of the $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$.