Let $X\in \operatorname{sSet}$ be a Kan complex. This gives rise to the fundamental groupoid $\Pi(X)$ of $X$. I am having trouble seeing why the composition in $\Pi(X)$ is well defined.
If $y\in X[2]$ is a 2-simplex, then the composition $d_0(y)\circ d_2(y)$ is defined to be $d_1(y)$. So if $y'\in X[2]$ is another 2-simplex with $d_0(y)=d_0(y')$ and $d_2(y)=d_2(y')$, then why do we have $d_1(y)=d_1(y')$ in $\Pi(X)$?
Clearly we need to define a horn $\Lambda^n\to X$ and use the Kan complex property to fill it to get a $2$-simplex in $X$ which gives $d_1(y)=d_1(y')$ but I just can't figure out which horn does the job.
If $\sigma$ and $\tau$ are $2$-simplices sharing their $1$-horn, say $x\xrightarrow{\alpha} y\xrightarrow{\beta} z$, then you may glue them together to a $1$-horn in a $3$-simplex by taking $\sigma$ as the $3$-face, $\tau$ as the $2$-face, and $s_1(\beta)$ as the $0$-face. The $1$-face of any $3$-simplex extending this horn then constitutes a homotopy between $d_2(\sigma)$ and $d_2(\tau)$.
Note that this uses inner horn fillings only and works for any $\infty$-category, see Proposition 1.3.2.7 in HTT.