For any $C^{\infty}$ manifold $M$, the tangent bundle $TM$ of $M$ is also a $C^{\infty}$ manifold.
Hence we can think about the differential $df:TM\rightarrow TN$ of maps $f:M\rightarrow N$ between smooth manifolds as a smooth map between two smooth manifolds $TM$ and $TN$.
When I come up with this, I'm totally convinced that for any (smooth) embedding $\iota:N\rightarrow M$ between smooth manifolds, the differential $d\iota:TN \rightarrow TM$ also becomes an embedding (It must be in common sense; embedding is subobject!), but I cannot prove this in detail.
In particular, I'm stuck on thinking about the differential of the differential $d(d\iota)$. It's quite hard to imagine for me.
Is this true? In that case, I want to see a detailed proof for this one.