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, 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.