Let $(X,\mathcal O\_X)$ be a ringed space (if necessary, assume that it is a scheme with suitable hypotheses). Given two complexes of sheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O\_X$-modules, one has the *complex of morphisms* \begin{equation} \operatorname{Hom}^\bullet(\mathcal F, \mathcal G),\end{equation}
and moreover the *complex of sheaves of morphisms* \begin{equation} \mathcal H om^\bullet (\mathcal F, \mathcal G), \end{equation}
which is defined on any open $U \subseteq X$ by $\mathcal H om^\bullet (\mathcal F, \mathcal G)(U) = \operatorname{Hom}^\bullet (\mathcal F\_{|U}, \mathcal G\_{|U})$.


We give the following definition: 



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> A complex of sheaves of $\mathcal O\_X$-modules $\mathcal I$ is called *K-injective* (or *hoinjective*) if for any exact (acyclic) complex of sheaves of $\mathcal O\_X$-modules $\mathcal A$, the complex $\operatorname{Hom}^\bullet(\mathcal A, \mathcal I)$ is acyclic.
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This definition should be precisely the same as the one I can found in the literature, for example [here](http://stacks.math.columbia.edu/tag/070G). My question is if the following characterisation holds:



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> A complex of $\mathcal O\_X$-modules $\mathcal I$ is K-injective if and only if for any acyclic complex of $\mathcal O\_X$-modules $\mathcal A$, the complex of sheaves $\mathcal H om^{\bullet}(\mathcal A, \mathcal I)$ is acyclic.
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One implication is trivial because $\operatorname{Hom}^\bullet(\mathcal F, \mathcal G) = \mathcal H om^\bullet (\mathcal F, \mathcal G)(X)$ by definition, but I'm confused about the other one.



K-injective (also known as hoinjective) complexes of sheaves of modules
Let $j:U\hookrightarrow X$ be the inclusion of an open subset. Then $\mathcal H om\_{\mathcal O\_X}^\bullet (\mathcal A, \mathcal I)(U) = \operatorname{Hom}^\bullet\_{\mathcal O\_U} (j^\ast\mathcal A,j^\ast\mathcal I)=\operatorname{Hom}^\bullet\_{\mathcal O\_X}(j\_!j^\ast\mathcal A,\mathcal I)$. The functors $j\_!$ and $j^\ast$ are exact, so they send acyclic complexes of sheaves to acyclic complexes of sheaves. Also $j\_!$ sends $\mathcal O\_U$-modules to $\mathcal O\_X$-modules. So, $j\_!j^\ast\mathcal A$ is an acyclic $\mathcal O\_X$-module. This gives a positive answer to your question.


