Base point of BU/BO, classifying space of U/O.

Question

There are principal bundles $$U(n) \to V_k(\mathbb{C}^n) \to G_k(\mathbb{C}^n)$$ and $$O(n) \to V_k(\mathbb{R}^n) \to G_k(\mathbb{R}^n),$$ where $V_k(\mathbb{F}^n)$ and $G_k(\mathbb{F}^n)$ are the complex/real ($\mathbb{F}=\mathbb{C},\mathbb{R}$) Stiefel and Grassmannian Manifold respectively. Customarily the universal bundles and the classifying spaces of $U=\operatorname{colim}_{\to n}U(n)$ or $O=\operatorname{colim}_{\to n}O(n)$ are now formed by passing to the inductive limit/colimit over $n$ and then $k$. Since homotopy classes of pointed maps are important for the classification I wondered what the base point of these spaces is? Based on the nlab article on the category of pointed objects (there is an error in 2., i.e. it should read "the colimits are the colimit in $\mathcal{C}$ of the diagrams with the basepoint adjoined" right?) it is "just" an adjoint point?