More generally, let $X_\cdot$ be a simplicial presheaf. As such, we can consider it as a simplicial object in presheaves, which in particular may be thought of as a simplicial object in simplicial presheaves $X'_\cdot.$ So we have:
$$X_\cdot:\Delta^{op} \to Set^{C^{op}}$$ and $$X'_\cdot =\left( \mspace{3mu} \cdot \mspace{3mu}\right)^{(id)} \circ X_\cdot:\Delta^{op} \to Set_{\Delta}^{C^{op}}$$ where $$\left( \mspace{3mu} \cdot \mspace{3mu}\right)^{(id)}:Set^{C^{op}} \to Set_{\Delta}^{C^{op}}$$ is the evident inclusion of presheaves into simplicial presheaves.
The homotopy colimit of $X'_\cdot$ in simplicial presheaves is computed "object-wise". Hence, for all $c \in C$ we have $$holim\left(X'\right)(C)=holim hocolim\left(X'\right)(C)=hocolim \left( X'\left(C\right)\right).$$ The right-hand side is the homotopy colimit of a simplicial object in simplicial sets, and can be computed by taking the diagonal of the corresponding bisimplicial set. But the diagonal of $X'\left(C\right)_\cdot$ is simply $X\left(C\right)_\cdot$ since $X'_\cdot$ in constant in one simplicial direction. Hence $$holim\left(X'\right)=X.$$$hocolim\left(X'\right)=X.$$1 More generally, let X_\cdot be a simplicial presheaf. As such, we can consider it as a simplicial object in presheaves, which in particular may be thought of as a simplicial object in simplicial presheaves X'_\cdot. So we have:$$X_\cdot:\Delta^{op} \to Set^{C^{op}}$$and $$X'_\cdot =\left( \mspace{3mu} \cdot \mspace{3mu}\right)^{(id)} \circ X_\cdot:\Delta^{op} \to Set_{\Delta}^{C^{op}}$$ where$$\left( \mspace{3mu} \cdot \mspace{3mu}\right)^{(id)}:Set^{C^{op}} \to Set_{\Delta}^{C^{op}}$$is the evident inclusion of presheaves into simplicial presheaves. The homotopy colimit of X'_\cdot in simplicial presheaves is computed "object-wise". Hence, for all c \in C we have$$holim\left(X'\right)(C)=holim \left( X'\left(C\right)\right).$$The right-hand side is the homotopy colimit of a simplicial object in simplicial sets, and can be computed by taking the diagonal of the corresponding bisimplicial set. But the diagonal of X'\left(C\right)_\cdot is simply X\left(C\right)_\cdot since X'_\cdot in constant in one simplicial direction. Hence$$holim\left(X'\right)=X.$\$