  For a pair $(H_1,H_2)$ of graphs, Lov\'{a}sz introduced a polytopal complex
called the Hom complex $\text{Hom}(H_1,H_2)$, in order to estimate topological
lower bounds for chromatic numbers of graphs. The definition is generalized to
hypergraphs. Denoted by $K_r^r$ the complete $r$-graph on $r$ vertices. Given
an $r$-graph $H$, we compare $\text{Hom}(K_r^r,H)$ with the box complex
$\mathsf{B}_{\text{edge}}(H)$, invented by Alon, Frankl and Lov\'{a}sz. We
verify that $\text{Hom}(K_r^r,H)$ and $\mathsf{B}_{\text{edge}}(H)$, both are
equipped with right actions of the symmetric group on $r$ letters $S_r$, are of
the same simple $S_r$-homotopy type.
