Category of graded modules

Let $(𝔄, +)$ be a monoid and $𝑅$ be a $𝔄$ -graded ring. The category of graded modules $𝖦 𝗋 𝔄 𝑅 𝖬 𝗈 𝖽$ is a category where an object is a $𝔄$ -graded module and a morphism is a $𝔄$ -graded $𝑅$ -module homomorphism. module