The question is: Prove that an equivalence between two additive categories is an additive functor.

By an additive category I mean a category with zero object, that every Hom-set of morphisms is a abelian group for sum, bilinear and every pair of objects have products.

Is this statement also true for adjoint functors?

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    $\begingroup$ See qchu.wordpress.com/2012/09/14/… for some discussion of the key fact that a functor between additive categories is additive iff it preserves biproducts iff it preserves coproducts iff it preserves products. $\endgroup$ – Qiaochu Yuan Oct 26 '14 at 5:18
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    $\begingroup$ Moreover the adjunction itself is additive, in the sense that one has an enriched natural isomorphism of hom-groups $\mathrm{Hom} (F X, Y) \cong \mathrm{Hom} (X, G Y)$. $\endgroup$ – Zhen Lin Oct 26 '14 at 8:08

Yes this is true. Specifically:

Proposition. Let $L:\mathcal B\to\mathcal A$ be left adjoint to $R:\mathcal A\to\mathcal B$ where $\mathcal A$ and $\mathcal B$ are additive categories. Then $L$ and $R$ are additive.

Proof. Recall that $F:\mathcal A\to\mathcal B$ is additive if and only if $F$ preserves binary products (prove this!). But $R$ is right adjoint so $R$ preserves limits. In particular, $R$ preserves binary products.

Proving $L$ is additive is similar once combined with the observation that binary products coincide with binary coproducts in an additive category. $\Box$

Of course, if you don't like using continuity properties of adjoint functors, you could directly prove the proposition. This is a nice exercise too.

Side-note. This is a useful result in geometry as a morphism $f:X\to Y$ of spaces defines functors $\DeclareMathOperator{Sh}{Sh}$ \begin{align*} f_* &:\Sh(X)\to\Sh(Y) \\ f^{-1} &: \Sh(Y)\to\Sh(X) \end{align*} One shows that $f^{-1}$ is left adjoint to $f_*$ and the above ensures the additivity of both.

In fact, one may further prove that right adjoint functors between abelian categories are left exact and left adjoint functors between abelian categories are right exact.

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    $\begingroup$ Left exactness can be defined for functors between arbitrary categories; it means that finite limits are preserved. Thus, right adjoint functors between arbitrary categories are left exact. For example, $\hom(X,-)$ is left exact for objects $X$ of any category. (In fact, this functor preserves arbitrary limits, which one calls continuous in analogy to analysis.) $\endgroup$ – Martin Brandenburg Oct 26 '14 at 8:42

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