Let ${\bf A}$ be a category. Let $\Delta[n]$ be the representable presheaf $\Delta(-,[n])$.

What is the colimit $$ \int^{n\in\Delta}\Delta[n]\times \coprod_{X_0,\dots X_n} {\bf A}(A,X_0)\times {\bf A}(X_0, X_1)\times\cdots \times {\bf A}(X_n,B) $$ What if ${\bf A}$ is simplicial and each ${\bf A}(X,Y)$ is a simplicial set, and the operation above lives in $\bf sSet$? Are there explicit descriptions of this object?