Definition Let $(X,x)$ and $(Y,y)$ be simplicial sets, and suppose we are given a pair of pointed maps $f,g: X \rightarrow Y$, which we identify with vertices of the simplicial set $\operatorname{Fun}( X, Y ) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} $. We will say that $f$ and $g$ are pointed homotopic if they belong to the same connected component of $\operatorname{Fun}( X, Y ) \times _{ \operatorname{Fun}( \{ x\} , Y) } \{ y\} $ (Definition