Remark 4.7.6.2. In the situation of Definition 4.7.6.1, two diagrams $f_0, f_1: B \rightarrow \operatorname{\mathcal{C}}$ are isomorphic relative to $A$ if and only if they satisfy the following pair of conditions:
The diagrams $f_0$ and $f_1$ have the same restriction to $A$: that is, we have $f_0|_{A} = \overline{f} = f_1|_{A}$ for some diagram $\overline{f}: A \rightarrow \operatorname{\mathcal{C}}$.
The diagrams $f_0$ and $f_1$ are isomorphic when viewed as objects of the $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(A,\operatorname{\mathcal{C}}) } \{ \overline{f} \} $.