I asked the same question a week ago on Mathematics Stackexchange but got no answer.

What would be a simple example of an additive functor $F:\mathcal C\to\mathcal C'$ of abelian categories such that the right derived functor
$$
RF:\text D(\mathcal C)\to\text D(\mathcal C')
$$
does **not** exist?

My reference for the notions involved in this post is the book **Categories and Sheaves** by Kashiwara and Schapira.

Here is a reminder of the definition of a right derived functor $RF$ in the above setting:

Let $\text K(F):\text K(\mathcal C)\to\text K(\mathcal C')$ be the triangulated functor induced by $F$ between the homotopy categories, let $X$ be in $\text K(\mathcal C)$, assume that the colimit
$$
\operatorname*{colim}_{X\to Y}\ \text K(F)(Y),\tag1
$$
where $X\to Y$ runs over all the quasi-isomorphisms out of $X$ in $\text K(\mathcal C)$, exists in $\text D(\mathcal C')$, and denote this colimit by $RF(X)$. The right derived functor $RF$ of $F$ is *defined at* $X\in\text D(\mathcal C)$ if, for any functor $G:\text D(\mathcal C')\to\mathcal A$, the colimit
$$
\operatorname*{colim}_{X\to Y}\ G(\text K(F)(Y))
$$
exists in $\mathcal A$ and coincides with $G(RF(X))$. The right derived functor $RF$ of $F$ *exists* if $RF$ is defined at $X$ for all $X$ in $\text D(\mathcal C)$.

The ideal would be to have an example of a pair $(F,X)$, where $F:\mathcal C\to\mathcal C'$ is an additive functor of abelian categories and $X$ is an object of $\text D(\mathcal C)$, such that (1) does **not** exist in $\text D(\mathcal C')$.