Let \(\mathcal D\) be a braided monoidal \(1\)-category. Assume \(\mathcal C_1\) is a monoidal \(1\)-category, and \(\mathcal C_2\) is a monoidal category over \(\mathcal D\), i.e. equipped with a monoidal functor \(g\colon \mathcal C_2\rightarrow\mathcal D\). Let now \(F\colon \mathcal C_1 \rightarrow\mathcal C_2\) be a monoidal functor. Then we can consider \(C_1\) as a monoidal \(1\)-category over \(\mathcal D\), namely with respect to \(f \coloneqq g\circ F \colon \mathcal C_1 \rightarrow\mathcal D\), and \(F\) becomes a monoidal functor over \(\mathcal D\).
A prebraiding over \(\mathcal D\) on \(F \colon \mathcal C_1 \rightarrow\mathcal C_2\) is then defined to be a prebraiding on \(F\) as in Definition 2.4.1, satisfying the additional condition that \(g\) maps the prebraiding isomorphisms in \(\mathcal C_2\) to the given braiding isomorphisms in \(\mathcal D\).