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 the function $\pi_0(i^W_{cod}\,):\pi_0(B)\to \pi_0(W)$ 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 the function $\pi_0(i^W_{dom}):\pi_0(A)\to \pi_0(W)$ is an injection.

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?

(Just for symmetry....)

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

Let $\mathcal{R'}\subset Mor_{1-Cob}$ 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, $\not\exists (S^1\hookrightarrow W)$.

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.

Edit provenance: In an earlier version of this question, I had reversed $\mathcal{L}$ and $\mathcal{R}$. This allowed Chris Schommer-Pries to correctly answer my question as originally posed, but not the question I meant to ask (i.e., the one you see above now).