  Let $F\colon \mathcal{C} \to \mathcal{E}$ be a functor from a category
$\mathcal{C}$ to a homological (Borceux-Bourn) or semi-abelian
(Janelidze-M\'arki-Tholen) category $\mathcal{E}$. We investigate conditions
under which the homology of an object $X$ in $\mathcal{C}$ with coefficients in
the functor $F$, defined via projective resolutions in $\mathcal{C}$, remains
independent of the chosen resolution. Consequently, the left derived functors
of $F$ can be constructed analogously to the classical abelian case.
  Our approach extends the concept of chain homotopy to a non-additive setting
using the technique of imaginary morphisms. Specifically, we utilize the
approximate subtractions of Bourn-Janelidze, originally introduced in the
context of subtractive categories. This method is applicable when $\mathcal{C}$
is a pointed regular category with finite coproducts and enough projectives,
provided these projectives are closed under protosplit subobjects, a new
condition introduced in this article and naturally satisfied in the abelian
context. We further assume that the functor $F$ meets certain exactness
conditions: for instance, it may be protoadditive and preserve proper morphisms
and binary coproducts - conditions that amount to additivity when $\mathcal{C}$
and $\mathcal{E}$ are abelian categories.
  Within this framework, we develop a basic theory of derived functors, compare
it with the simplicial approach, and provide several examples.
