Let $C$ be a cocomplete category and $Simp(C)=C^{\triangle^{op}}$ the category of simplicial objects in $C$. I want to show that $Simp(C)$ is simplicially enriched but I don't understand how the enrichemt supposed to look. Here's what I tried:
Let $X_{\bullet},Y_{\bullet} \in Simp(C)$. The end formula gives $Nat(X_{\bullet},Y_{\bullet})= \int_{n \in \triangle^{op}} C(X_n,Y_n)$. But I have no idea why this is a simplicial set. It looks like a set to me.
The standard simplicial enrichment on $Simp(C)$ is defined as follows. First, note that since $C$ is cocomplete, it can be considered to be tensored over $Set$: If $X$ is an object of $C$ and $S$ is a set, then $X\otimes S$ is a coproduct of copies of $X$ indexed by elements of the set $S$. If $K$ is a simplicial set and $A$ is an object of $Simp(C)$, we can define now define an object $A\otimes K$ of $Simp(C)$ by $$(A\otimes K)_n=A_n\otimes K_n,$$ where $A_n\otimes K_n$ is defined as above (the coproduct of copies of $A_n$ indexed by elements of $K_n$). The structure maps of the simplicial object $A\otimes K$ are defined in the obvious way (a map $n\to m$ in $\Delta$ gives maps $A_m\to A_n$ and $K_m\to K_n$, which induce a map $A_m\otimes K_m\to A_n\otimes K_n$).
This tensor operation can then be used to define a simplicially enriched category whose objects are the objects of $Simp(C)$: given two such objects $A$ and $B$, the Hom-simplicial set $\mathbf{Hom}(A,B)$ is defined by $$\mathbf{Hom}(A,B)_n=\operatorname{Nat}(A\otimes \Delta^n,B).$$
Of course, there are various details to be checked here to verify that this all makes sense and really gives a tensored simplicially enriched category, but they should all be pretty straightforward.