Let $\mathscr{C}$ be a category which admits enough injectives and let $\mathscr{I}$ be the full subcategory of injective objects.
Let $F:\mathscr{C}\rightarrow\mathscr{C}'$ be an additive functor of abelian categories. In every textbook containing stuff about a derived category, we can find the theorem that states if $F$ is left exact, then it has a right derived functor.
One of the key points in the proof is that $K^{+}F:K^{+}(\mathscr{C})\rightarrow K^{+}(\mathscr{C}')$ sends acyclic complexes of injective objects to acyclic complexes (since the functor is additive and every short exact sequence of injective objects splits), and here is where the left exactness of $F$ is used.
Now the weird thing is that I have an easier argument in mind that does not use the left exactness condition, namely:
Let $F:\mathscr{C}\rightarrow\mathscr{C}'$ an additive functor. Then $\mathcal{F}:K^{+}(\mathscr{C})\rightarrow K^{+}(\mathscr{C}')$ sends acyclic complexes of injective objects to acyclic complexes.
Proof:
The acyclic complexes of injective objects are all homotopy equivalent to the zero complex and additive functors send the zero object to the zero object (since it send the identity on the zero object to the identity and zero morphisms to zero morphisms and the zero object in a category is characterized by the fact that those two coincide), so the image of an acyclic complex is homotopy equivalent to zero in $K^{+}(\mathscr{C}')$ and hence acyclic.
Either this argument is wrong, or the whole mathematical community seems to stick to the particular assumption of left exactness for no reason whatsoever, but I am modestly assuming the first. Could anyone enlighten me?