Remark 7.6.5.17. Let $F_0, F_1: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, which we identify with a diagram $\sigma : ( \bullet \rightrightarrows \bullet ) \rightarrow \operatorname{\mathcal{QC}}$. Unwinding the definitions, we see that extensions of $\sigma $ to a diagram $\overline{\sigma }: ( \bullet \rightrightarrows \bullet )^{\triangleleft } \rightarrow \operatorname{\mathcal{QC}}$ can be identified with the following data:
An $\infty $-category $\operatorname{\mathcal{E}}$ equipped with functors $G: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $H: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.
Isomorphisms $\alpha _0: F_0 \circ G \xrightarrow {\sim } H$ and $\alpha _1: F_1 \circ G \xrightarrow {\sim } H$ in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{C}})$.
In this case, we can identify the quadruple $(G, H, \alpha _0, \alpha _1)$ with a single functor of $\infty $-categories