Remark 3.1.3.11. 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 the simplicial set $\operatorname{Fun}_{A/ \, /S}( B, X)$ can be identified with the fiber of the induced map
\[ \operatorname{Fun}(B,X) \rightarrow \operatorname{Fun}(A,X) \times _{ \operatorname{Fun}(A,S) } \operatorname{Fun}(B,S) \]
over the vertex given by the pair $(f,g)$.