TITLE: Meaning of functorial
QUESTION [5 upvotes]: It's known that for a short exact sequence of complexes, $0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$, it associates a homology sequences $...\rightarrow H(E')\rightarrow H(E)\rightarrow H(E'')\rightarrow...$ with the connecting homomorphism $\delta$.
On page 769 of Lang's Algebra, it says

The map $\delta$ is functorial (in obvious sense), and hence the the structure $(H,\delta)$ is a functor from the category of short exact sequences of complexes into the category of complexes.

What does it mean by functorial exactly?

REPLY [4 votes]: Given a morphism of short exact sequences, you get a morphism of long exact sequences, and this map respects composition. $\delta$ itself is, most precisely, a natural transformation between the functors $H_n\circ t$ and $H_{n-1}\circ f$ from short exact sequences of chain complexes to abelian groups, where $t$ sends an s.e.s. to its third complex and $f$, to its first. This just says that the squares involving $\delta$ and a morphism of short exact sequences of chain complexes commute.