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Let a class of objects be enriched in the category of abelian groups, i.e. for each ordered pair of objects, an abelian group is done, with an operation $+$, identities and compositions are morphisms coming from the monoidal structure on $\operatorname{Ab}$ given by tensor product of $\mathbb{Z}$-modules and satisfy diagrams for associativity and unity.

My question is: is it always true in such a context that $(f+g)\circ h=f\circ h+g\circ h$ and $k\circ (f+g)=k\circ f +k\circ g$? In other words, monoidal structure is enough to ensure bilinearity of composition or I need to assume it?

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  • $\begingroup$ This follows from the universal property of the tensor product in the category of abelian groups. $\endgroup$ – asdq May 22 '18 at 10:13
  • $\begingroup$ @asdq I am not able to understand what tensor product properties have to do with composition of morphisms' properties. $\endgroup$ – bateman May 22 '18 at 10:15
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    $\begingroup$ This is assumed in the definition of enriched category, as $\circ : hom(b,c)\otimes hom(a,b) \to hom(a,c)$ is a morphism in the monoidal category over which you are enriched $\endgroup$ – Max May 22 '18 at 10:33
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Given $f: A \to B$, $g: A \to B$ and $h:A' \to A$, the expression $(f+g)\circ h$ is a notation for $c_{A,B,C}((f+g)\otimes h)$ where $$ c_{A,B,C} : \hom(A,B) \otimes \hom(A',A) \to \hom(A',B) $$ is the morphism of composition. As you said, it is a morphism of abelian groups, and in the category of abelian groups morphisms $X\otimes Y \to Z$ corresponds bi-univocally to bilinear maps $X\times Y\to Z$. More precisely there is a bilinear map $\eta : X\times Y \to X\otimes Y$ such that for every bilinear $f: X\times Y \to Z$ there exists a unique $\hat f: X \otimes Y \to Z$ such that $\hat f \eta = f$. In particular, you can see that $$\hat f ((x+x')\otimes y) = \hat f(x\otimes y) + \hat f(x'\otimes y)$$

Coming back to your problem, we have that $$ c_{A,B,C} ((f+g)\otimes h) = c_{A,B,C} (f\otimes h) + c_{A,B,C} (g\otimes h) $$ But the latter is also denoted $f \circ h + g \circ h$.

The same goes for the other linearity, so in the end bilinearity of composition is part of the definition of an $\mathbf{Ab}$-enriched category.

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Having an $\rm{Ab}$-enrichment means that for any two objects $a,b$ we have $\hom(a,b)\in \rm{Ab}$ and composition is given by a map $\hom(a,b)\otimes\hom(b,c)\to \hom(a,c)$ in $\rm{Ab}$. But this means precisely that we have a bilinear map $\hom(a,b)\times \hom(b,c)\to \hom(a,c)$.

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