Let $1-Cob$ denote the category of oriented 0-manifolds and oriented cobordisms between them. If $W:A\to B$ is a cobordism, i.e. $\partial W\cong A+B$, we write $i^W_{dom}:A\to W$ and $i^W_{cod}:B\to W$ to denote the boundary-component inclusions. Let $\pi_0:{\bf Man}\to{\bf Set}$ denote the connected components functor.

Let $\mathcal{L}\subset Mor_{1-Cob}$ denote the set of morphisms $W$ for which $\pi_0(i^W_{dom})$ is an injection. 

Let $\mathcal{R}\subset Mor_{1-Cob}$ denote the set of morphisms $W$ for which $\pi_0(i^W_{cod})$ is an injection and for which the 1-manifold $W$ contains no closed loops, $\not\exists (S^1\hookrightarrow W)$.

I believe that $\mathcal{L}$ and $\mathcal{R}$ form an orthogonal factorization system on $1-Cob$, but I don't know how to prove it. 

**Question**: Can you find a proof, a reference, or a counterexample for this conjecture?

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(Just for symmetry....)

Let $\mathcal{L'}\subset Mor_{1-Cob}$ denote the set of morphisms $W$ for which $\pi_0(i^W_{dom})$ is an injection and for which the 1-manifold $W$ contains no closed loops, $\not\exists (S^1\hookrightarrow W)$. 

Let $\mathcal{R'}\subset Mor_{1-Cob}$ denote the set of morphisms $W$ for which $\pi_0(i^W_{cod})$ is an injection.

I also believe that $\mathcal{L'}$ and $\mathcal{R'}$ form an orthogonal factorization system on $1-Cob$, but I don't know how to prove it either.