The first Cech derived functor

Let $\mathcal{C}$ be a semi-abelian category, $\mathcal{D}$ a Birkhoff subcategory of $\mathcal{C}$, and $I$ the associated reflector. For $C\in \mathcal{C}$, we establish an isomorphism between ${\mathrm{H}}_1(\mathrm{N}(I(\check{C}(f,C))))$ (where $\mathrm{N}$ is the Moore normalization functor and $\check{C}(f,C)$ is a Cech resolution associated to $C$) and ${\mathrm{H}}_1(\mathrm{N}(I(\mathbb{G}(C))))$ (where $\mathbb{G}(C)$ is the canonical $\mathbb{G}$-simplicial resolution of $C$ for $\mathbb{G}$ the comonad derived from an adjunction $F\text{\hspace{0.17em}⁣}:\mathcal{X}\to \mathcal{C}⊣U\text{\hspace{0.17em}⁣}:\mathcal{C}\to \mathcal{X}$ satisfying classical conditions introduced by Quillen). The former corresponds to the Hopf formula for the second homology object associated to $I$, while the latter represents the first \v Cech derived functor of $I$. Specifically, we show that both can be expressed by a single unified formula. Furthermore, we can also prove that, under certain assumptions, the first \v Cech derived functor of $I$ is naturally isomorphic to the first non-additive derived functor of $I$ (in the sense of Culot–Renaud–Van~der~Linden).