Let $M_1\subset M_2$ and $K_1\subset K_2$ be inclusions of simplicial monoids. If there is a weak equivalence $f:M_2\to K_2$ that restricts to a weak equivalence $f|_{M_1}:M_1\to K_1$, does this induce a weak equivalence on the cokernels ${\mathrm{coker}}(M_1\to M_2)\to {\mathrm{coker}}(K_1\to K_2)$?

I know that homotopy colimits respects weak equivalence, but for the special case of the cokernel, can we just use the ordinary colimit?

**Edit:** Here is my actual situation. Let $A$ be a pointed simplicial subset of $X$. Consider the map $J(A)\to J(X)$ that sends $a\mapsto a^2$ where $J$ is James construction (i.e. the reduced free monoid construction). I trying to show that the group reflection ${\mathrm{coker}}(J(A)\to J(X))\to {\mathrm{coker}}(F(A)\to F(X))$ is a weak equivalence (or at least induces isomorphisms in homology). Here $F$ is Milnor's construction (i.e. the reduced free group construction).

The problem I'm facing is that map $a\mapsto a^2$ is not induced by a simplicial map $A\to X$. Otherwise, given a simplicial map $f:A\to X$, then ${\mathrm{coker}}(J(A)\xrightarrow{J(f)} J(X))= J({\mathrm{coker}}(A\xrightarrow{f} X))$ and we are done.