Homomorphism of graded vector spaces

Let $𝑓 : 𝑉 \to 𝑊$ be a linear map of $𝑆$ -graded vector spaces over $𝕂$ .

$𝑓$ is graded or grade-preserving iff $𝑓 (𝑉 𝛼) \leq 𝑓 (𝑊 𝛼)$ for all $𝛼 \in 𝑆$ . linalg
See Strict category of graded vector spaces.
If $(𝑆, +)$ is a monoid, then $𝑓$ is homogenous of degree $𝛽 \in 𝑆$ iff $𝑓 (𝑉 𝛼) \leq 𝑓 (𝑊 𝛼 + 𝛽)$ and we write $d e g 𝑓 = 𝛽$ . linalg
See Closed category of graded vector spaces