If I have $k$-vector spaces $U,V,W,X$, is every linear map $f:U\oplus V\to W\oplus X$ of the form $f_1 + f_2$ where $f_1:U\to W$ and $f_2:V\to X$? It is true if I consider $f_i$ having images in $W\oplus X$, both.
If I consider these direct sums as a $\mathbb{Z}_2$-grading, i.e. I have $f$ being a map of super-vector spaces, then this should be true, shouldn't it? (Because in the category of super-vector spaces we only consider even maps.)