Remark 3.1.3.8. Let $B$ and $X$ be simplicial sets, and let us identify vertices of $\operatorname{Fun}(B,X)$ with morphisms $\overline{f}: B \rightarrow X$ in the category of simplicial sets. Then:
Suppose we are given another simplicial set $A$ equipped with a pair of morphisms $i: A \rightarrow B$ and $f: A \rightarrow X$. Then vertices of the simplicial set $\operatorname{Fun}_{A/}(B,X)$ can be identified with morphisms $\overline{f}: B \rightarrow X$ satisfying $f = \overline{f} \circ i$.
Suppose we are given another simplicial set $S$ equipped with a pair of morphisms $g: B \rightarrow S$ and $q: X \rightarrow S$. Then vertices of the simplicial set $\operatorname{Fun}_{/S}(B,X)$ can be identified with morphisms $\overline{f}: B \rightarrow X$ satisfying $g = q \circ \overline{f}$.
Suppose we are given a square 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 vertices of the simplicial set $\operatorname{Fun}_{A/ \, /S}(B, X)$ can be identified with solutions of the associated lifting problem: that is, morphisms of simplicial sets $\overline{f}: B \rightarrow X$ satisfying $f = \overline{f} \circ i$ and $g = q \circ \overline{f}$.