Let $X$ be a topological space and let $PX$ be its space of paths. Let $I=[0,1]$ with coordinate $s$. There is an homotopy $$ F\: : \: I\times PM\to PM $$ Defined by $F(s,y)(t):=y(st)$. This map is an homotopy between $f_{0}\: : \: PX\to PX$ , $(f_{0}(y))(t)=y(0)$ and the identity. Let $K$ be a simplicial set. I would like to find a map between simplicial sets $$ F\: : \: \Delta[1]\times Map_{sSet}(\Delta[1],K )\to Map_{sSet}(\Delta[1],K ) $$ which is an homotopy between the identity and the "constant path at the initial point". Do you have some ideas? Do I use a fibrant simplicial set $K$?

Let $X$ be a topological space and let $PX$ be its space of paths. Let $I=[0,1]$ with coordinate $s$. There is an homotopy $$ F\: : \: I\times PM\to PM $$ Defined by $F(s,y)(t):=y(st)$. This map is an homotopy between $f_{0}\: : \: PX\to PX$ , $(f_{0}(y))(t)=y(0)$ and the identity. Let $K$ be a simplicial set. I would like to find a map between simplicial sets $$ F\: : \: \Delta[1]\times Map_{sSet}(\Delta[1],K )\to Map_{sSet}(\Delta[1],K ) $$ which is an homotopy between the identity and the "constant path at the initial point". Do you have some ideas? Do I use a fibrant simplicial set $K$? Is it correct to define the path space of a simplicial set $K$ with $ Map_{sSet}(\Delta[1],K )$?