Subgroup functor

Definition
Let $$F$$ be a group functor on a category $$C$$. A subgroup functor of $$F$$ is another group functor $$F_1$$ on $$C$$, along with, for each $$A \in C$$, an injective homomorphism $$i(A):F_1(A) \to F(A)$$, such that for $$f:A \to B$$:

$$F(f) \circ i(A) = i(B) \circ F_1(f)$$

Intuitively it associates, for each $$A$$ a subgroup $$F_1(A)$$ of $$F(A)$$.