I am a beginner of $\infty$-categories and am trying to follow this video course. I now have some trouble with the exercise 2 appearing at around 30:20 of this video, which asks me to show that in $Sing(X),$ the mapping space $Map(a,b)$ is homotopy equivalent to the space of paths from $a$ to $b$ in $X.$ Here $Map(a,b)$ is defined as the pullback of $\Delta^0\xrightarrow{(a,b)}Sing(X)\times Sing(X)\leftarrow Fun(\Delta^1,Sing(X)),$ where the last term is simplicial mapping space and the second map is taking two vertices.
To make sure that I understand the question correctly, if we write $Z$ as the space of paths from a to b in $X,$ then I think $Z$ can be written as pullback of $*\xrightarrow{(a,b)} X\times X\leftarrow X^{[0,1]}$ and we need to show that $|Map(a,b)|$ is homotopy equivalent to $Z$. I know that $|\Delta^0|=*$ and there are weak homotopy equivalence $|Sing(X)\times Sing(X)|\xrightarrow{\sim} X\times X$ and $|Fun(\Delta^1,Sing(X))|\xrightarrow{\sim}|Sing(X)|^{[0,1]}$ as $Sing(X)$ is a Kan complex. (Do we have $|Sing(X)|^{[0,1]}$ weakly homotopy equivalent to $X^{[0,1]}$? ) But I don't see whether this can be of any help. Could somebody help me or provide another way to tackle?