What's the meaning of homomorphism preserving the gradings here?

Question

In the sentence labeled by red line, what's the meaning of homomorphism preserving the gradings here?

Answer 1

The fact that $S$ and $T$ are graded means that we have decompositions $S=\bigoplus S_d$ and $T= \bigoplus T_d$, and the fact that $\phi: S\to T$ preserves the grading means that $\phi(S_d)\subset T_d$ for any $d$.