Remark 3.1.5.14. We can describe the strict $2$-category $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ more informally as follows:
The objects of $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ are Kan complexes.
The morphisms of $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ are morphisms of Kan complexes $f: X \rightarrow Y$.
If $f_0, f_1: X \rightarrow Y$ are morphisms of Kan complexes, then a $2$-morphism $f_0 \Rightarrow f_1$ in $\mathrm{h}_{2} \mathit{\operatorname{Kan}}$ is an equivalence class of homotopies $h: \Delta ^1 \times X \rightarrow Y$ from $f_0 = h|_{\{ 0\} \times X}$ to $f_1 = h|_{\{ 1\} \times X}$, where we regard $h$ and $h'$ as equivalent if they are homotopic relative to $\operatorname{\partial \Delta }^1 \times X$.