Remark 3.1.3.10. Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{i} & X \ar [d]^{q} \\ B \ar [r]^-{g} \ar@ {-->}[ur]^{ \overline{f} } & S. } \]
Then:
If $S \simeq \Delta ^{0}$ is a final object of the category of simplicial sets, then we have an equality $\operatorname{Fun}_{A/ \, /S}(B,X) = \operatorname{Fun}_{A/}(B,X)$ (as simplicial subsets of $\operatorname{Fun}(B,X)$).
If $A \simeq \emptyset $ is an initial object of the category of simplicial sets, then we have an equality $\operatorname{Fun}_{A/ \, /S}(B,X) = \operatorname{Fun}_{/S}(B,X)$ (as simplicial subsets of $\operatorname{Fun}(B,X)$).
If $S \simeq \Delta ^{0}$ and $A \simeq \emptyset $ are final and initial objects, respectively, then we have an equality $\operatorname{Fun}_{A/ \, /S}(B,X) = \operatorname{Fun}(B,X)$.