Morphisms in the category $\text{Comp}$ are abelian groups?

Question

I'm reading Rotman's Introduction to Algebraic Topology and he states that the set of morphisms between two chain complexes in the category $\text{Comp}$ (defined in the box below) is in fact an abelian group.

$(S_*, \partial) \in \text{Obj Comp}$, where $(S_*, \partial)$ means chain complex.

Hom$(S_*', S_*) = \{ \text{Set of all chain maps between $S_*'$ and $S_*$}\}$

Composition is defined as: $\{f_n\} \circ \{g_n \} = \{f_n \circ g_n\}$, for chain maps $f=\{f_n\}$, and $g=\{g_n\}$.

He states that if $f=\{f_n\}$ and $g=\{g_n\} \in $ Hom $(S_*', S_*)$, then , $f + g \in$ Hom$(S_*', S_*)$. I'm trying to prove to myself that this is a group, but what does this operation represent? What is $\{f_n\} + \{g_n\}$?

Answer 1

$\{f_n\} + \{g_n\}$ is the family of morphism of abelian groups $S'_n \to S_n, x \mapsto f_n(x) + g_n(x)$.

Answer 2

$$\{ f_n \} + \{ g_n \} = \{ f_n + g_n \} $$

That is, the $n$-th component of the sum of chain morphisms is the sum of the $n$-th components of the two terms.

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If you've never seen the a sum of morphisms, then I will presume you are working in the special case of chain complexes of abelian groups.

Addition is a standard operation on abelian group homomorphisms, and is defined pointwise: if $u,v : G \to H$ are abelian group homomorphisms, then $u+v : G \to H$ is a group homomorphism as well, with values given by

$$ (u + v)(g) = u(g) + v(g) $$