How to directly show that Figure 8 injective immersion is not a monomorphism

Question

I'm working in the category of smooth manifolds. The injective immersion that takes the open unit interval $(0,1)$ to the figure 8 is a well-known example of an injective immersion that is not an embedding. Now embeddings are the monomorphisms in our category. So the figure 8 immersion (let's call it $f$) must not be a monomorphism. Hence, there must exist two distinct maps $g_1,g_2:X\rightarrow (0,1)$ such that $f\circ g_1 = f\circ g_2$. What is an example of such maps?

Answer 1

Let $f : X \to Y$ be any smooth map whatsoever. Suppose $f (x) = f (x')$. Then $x$ and $x'$, considered as smooth maps $1 \to X$, have the property that $f \circ x = f \circ x'$. So if $f$ is monic in the category of manifolds it must be injective.

Conversely, suppose $f : X \to Y$ is an injective smooth map. Then, for any smooth maps $g, g' : T \to X$, if $f \circ g = f \circ g'$, then $g = g'$ (because smooth maps are determined by their action on points). Thus any injective smooth map is monic. In particular the figure-8 injective immersion is monic.