The cobordism hypothesis is a conjectural characterization of the framed cobordism $(\infty,n)$-category as the free symmetric monoidal $(\infty,n)$-category with duals generated by a single object. After its original formulation by Baez and Dolan in 1995, a strategy for a proof of the conjecture was put forward by Lurie in 2009. Though this strategy is very efficient in reducing the general hypothesis to a relatively concrete statement (Claim 3.4.17 in Lurie's text), a formal proof of this concrete statement has yet to appear in the literature. In addition, this strategy does not cover the 1-dimensional case. In this talk I will describe a way to extend Lurie's strategy to the case of $n = 1$, in which case the analogue of the missing claim can be proved using, among other things, the notion of quasi-unital $\infty$-categories.