In topology, if a map $Y\to B$ is some sort of fibration then you would think that $B$ being equivalent to a subspace $A$ would imply that $Y$ is equivalent to $Y_A=Y\times_AB$. If fibration means Serre fibration and equivalent means weakly, then you might want to use homotopy groups and the Whitehead Theorem. But what if we try to prove directly that a deformation retraction of $B$ to $A$ would yield a deformation retraction of $Y$ to $Y_A$? Then we will want to assume that $Y\to B$ has the 'relative homotopy lifting property' for the pair $(Y,Y_A)$, that is, the right lifting property with respect to the inclusion $Y\times 0\cup Y_A\times I\to Y\times I$.
I suppose that this it how it is in what you are looking at: Any right fibration of simplicial sets has the RLP w.r.t. $K\times \lbrace 1\rbrace \cup L\times \Delta^1\to K\times \Delta^1$ for every $L\subset K$ (but not w.r.t. $K\times \lbrace 0\rbrace \cup L\times \Delta^1\to K\times \Delta^1$), and this is good for lifting the sort of deformation retraction that exists from a simplex to its zeroth vertex.
Something like that.