I can muddy the waters...!
According to editor E. Scholz of Hausdorff’s Collected Works (2008, p. 884):
In a note of 3/20/1933 (Nachlass, fasc. 449) and in a further undated note (fasc. 571), Hausdorff symbolized the functoriality property of homology (in our later terminology) with a commutative diagram of homomorphisms between the terms of two sequences of groups $(A_n)_{n\in\mathbf N}$, $(A'_n)_{n\in\mathbf N}$:
(Nachlass, fasc. 571, leaf 1).
Could this have, somehow, made its way out of Bonn (where Hausdorff lectured on combinatorial topology that year) and to Hurewicz, Eilenberg, Steenrod, et al.?