Monoidal natural transformation

In order to categorify the notion of monoid homomorphism it is necessary to impose data and axioms on both the notion of functor and of natural transformation between such functors. Let $𝑇 1, 𝑇 2 : 𝖢 \to 𝖣$ be monoidal functors with $(𝜖 1, 𝜇 1)$ and $(𝜖 2, 𝜇 2)$ respectively. A natural transformation $𝛾 : 𝑇 1 \Rightarrow 𝑇 2 : 𝖢 \to 𝖣$ is called monoidal iff cat