Given an exact category
$${\mathcal {C}}$$
, it is well known that the connected component reflector
$$ \pi _0 :\mathsf {Gpd}(\mathcal {C}) \rightarrow \mathcal {C}$$
from the category
$$\mathsf {Gpd}(\mathcal {C})$$
of internal groupoids in
$$\mathcal {C}$$
to the base category
$$\mathcal {C}$$
is semi-left-exact. In this article we investigate the existence of a monotone-light factorization system associated with this reflector. We show that, in general, there is no monotone-light factorization system
$$(\mathcal {E}',\mathcal {M}^*)$$
in
$$\mathsf {Gpd}$$
(
$$\mathcal {C}$$
), where
$$\mathcal {M}^*$$
is the class of coverings in the sense of the corresponding Galois theory. However, when restricting to the case where
$$\mathcal {C}$$
is an exact Mal’tsev category, we show that the so-called comprehensive factorization of regular epimorphisms in
$$\mathsf {Gpd}$$
(
$$\mathcal {C}$$
) is the relative monotone-light factorization system (in the sense of Chikhladze) in the category
$$\mathsf {Gpd}$$
(
$$\mathcal {C}$$
) corresponding to the connected component reflector, where
$$\mathcal {E}'$$
is the class of final functors and
$$ \mathcal {M}^*$$
the class of regular epimorphic discrete fibrations.