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Example (Simplicial Sets). Let $\operatorname{Set_{\Delta }}$ denote the category of simplicial sets. Then $\operatorname{Set_{\Delta }}$ can be regarded as (the underlying ordinary category of) a simplicial category, which we will also denote by $\operatorname{Set_{\Delta }}$: given a pair of simplicial sets $X_{\bullet }$ and $Y_{\bullet }$, we define $\operatorname{Hom}_{ \operatorname{Set_{\Delta }}}( X_{\bullet }, Y_{\bullet } )_{\bullet }$ to be the simplicial set $\operatorname{Fun}( X_{\bullet }, Y_{\bullet })$ parametrizing morphisms from $X_{\bullet }$ to $Y_{\bullet }$ (see Construction