If M a monoid and $Set^M$ a topos associate to M, I found $Set^M$ have (epi, strong,mono) factorization system(<https://math.stackexchange.com/questions/541300/epi-mono-factorization-in-presentable-categories>), I think $Set^M$ has (epi,mono source) factorization [[http://katmat.math.uni-bremen.de/acc/acc.pdf](http://katmat.math.uni-bremen.de/acc/acc.pdf), pag. 257] but I can not found the proof of that, I would like see some reference of the proof such that don't use the fact $Set^M$ is a locally presentable category or If I suppose it's wrong a liked see the counterexample.


Note: If $(f\_i, X)$ is a source and $f: X \rightarrow X/ \sim$ where $a \sim b$ iif for all $f\_i : X \rightarrow Y\_i$, $f\_i(a) = f\_i(b)$, the first I think was associate the action of $X$ to $X / \sim$ so $\lambda: X \times M \rightarrow X$, we have $f(\lambda(x,m))$ and $\lambda(f(x),m)$ but I think, I can´t write $f(\lambda(x,m)) = \lambda(f(x),m)$ and finished the proof, because I can't
justify the equality.



(Epi,mono source) factorization in M-Set
Based of the idea of [მამუკა ჯიბლაძე](https://mathoverflow.net/users/41291/%E1%83%9B%E1%83%90%E1%83%9B%E1%83%A3%E1%83%99%E1%83%90-%E1%83%AF%E1%83%98%E1%83%91%E1%83%9A%E1%83%90%E1%83%AB%E1%83%94) and $Set$ has (epi, mono source) factorization, if $(f\_i, (X, \lambda))$ is a source, the (epi, initial source factorization) is $(((X/ \sim), \lambda), f\_i)$ where $a \sim b$ iff $f\_i(a)=f\_i(b)$ for all $i \in I$ and $f: (X, \lambda) \rightarrow (X/\sim, \lambda)$ preserve the action because:
$$f\_i(\lambda(f(x),m)) = \mu( f\_i(f(x)),m)$$
$$= \mu( f\_i(x)),m)$$
$$= f\_i(\lambda(x,m))$$
$$= f\_i(f(\lambda(x,m)))$$
$$\lambda(f(x),m)) = f(\lambda(x,m))$$
and $f\_i: (X/\sim, \lambda) \rightarrow (Y\_i, \mu\_i)$.


