Let $B$ be an object equipped with two maps $ \mu: B \times B \to B$ and $\rho : B \to B$ in a cocomplete model category $\mathcal{M}$ such that $(B, \mu, \rho )$ is a group object. These two maps induce two maps $$\mu_*: \mathcal{M}(A, B) \times \mathcal{M}(A, B) \to \mathcal{M}(A,B)$$ and $$\rho _* : \mathcal{M}(A, B) \to \mathcal{M}(A,B)$$ given by $\mu_*(f, g)(a) = \mu(f(a), g(a))$ and $\rho _*(f)(a) = \rho (f(a)).$
We define a relation $\sim$ on $\mathcal{M}(A, B)$ by $f \sim g$ if and only if there exists $F \in \mathcal{M}(CA, B)$ such that $$\mu_*(\rho _*(f), g) = i^*(F)$$ where $i^*$ is the map $\mathcal{M}(CA, B) \to \mathcal{M}(A, B)$ induces by $i : A \to CA.$
Now, my questions are the following:
1) Is $\sim$ an equivalence relation?
2) Does $\mu_*$ induce a group structure on $\mathcal{M}(A, B)/ \sim?$
Any help will be appreciated.
Thank you in advance.