Lemma 14.26.10. Let $\mathcal{C}$ be a category.

If $a_ t, b_ t : X_ t \to Y_ t$, $t \in T$ are homotopic morphisms between simplicial objects of $\mathcal{C}$, then $\prod a_ t, \prod b_ t : \prod X_ t \to \prod Y_ t$ are homotopic morphisms between simplicial objects of $\mathcal{C}$, provided $\prod X_ t$ and $\prod Y_ t$ exist in $\text{Simp}(\mathcal{C})$.

If $(X_ t, Y_ t)$, $t \in T$ are homotopy equivalent pairs of simplicial objects of $\mathcal{C}$, then $\prod X_ t$ and $\prod Y_ t$ are homotopy equivalent pairs of simplicial objects of $\mathcal{C}$, provided $\prod X_ t$ and $\prod Y_ t$ exist in $\text{Simp}(\mathcal{C})$.

## Comments (0)