Determinantal/Fitting ideal of the direct sum of linear maps

5.6. Fact. If G and H are matrices such that GH is defined, then for all n≥0, we have $D_n(GH)⊆D_n(G)D_n(H)$
Proof: By considering exterior powers, we reduce to the n=1 case, for which it is clear.
The following equality is immediate $D_n(varphi ⊕ psi)=sum_{k=0}^{n}D_k(varphi)D_{n-k}(psi)$

In the above, $D_n$ is a determinantal ideal and $psi$ and $varphi$ are linear maps over free modules of finite rank.
I feel like I’m being silly, but I can’t see how it is immediate. Any help would be appreciated.
To be clear, I’m asking how the last equation follows from just the theorem and other elementary facts about determinantal ideals. I would also be happy to see other constructive proofs of this equation.

In particular I imagine that we should have to use the elementary fact that if you have a block matrix $G$ with one block the identity of size r and the other block the matrix $H$, then $D_{k+r}(G)=D_k(H)$ for every r≥0
Since the direct sum of matrices can be decomposed into the product of two such matrices, I think this will be used to deduce the result, but I cannot see how.