Let $M,N$ be smooth manifolds and $f: M \to N$ a smooth map. Then, we have the following commutative diagram
Here, $\pi_1, \pi_2$ are the bundle projections from the tangent bundles, $p_1, p_2$ are the first and second projections from the pullback bundle, $Df$ is the derivative of $f$ and $f_*$ is the morphism induced by the universal property of pullback bundles. The notes that I am following for these topics claims that when $f$ is an immersion, $TM$ is a sub-bundle of $f^*TN$ and when $f$ is a submersion, $f^*TN$ is a quotient bundle of $TM$.
The claim is clear when $f$ is an immersion, as in that case $Df$ is injective everywhere. However, I have no idea how to prove the claim for the case when $f$ is a submersion. If it could be shown that $f_*$ is a constant rank morphism of bundles, maybe the kernel of the morphism would lead to a quotient? But I'm not clear how to check all the details even then. Any ideas on how to proceed is greatly appreciated.
Fix a point $p \in M$. By definition of submersion, the differential $(Df)_{p}\colon (\operatorname{T}M)_{p} \to (\operatorname{T}N)_{p}$ is a surjection. By the construction of the pullback bundle, $p_{2}$ is fibre-wise an isomorphism. In particular, $(p_{2})_{p}\colon (f^{*}\operatorname{T}N)_{p} \to (\operatorname{T}N)_{p}$ is an isomorphism. The map $f_{*,p}\colon (\operatorname{T}M)_{p} \to (f^{*}\operatorname{T}N)_{p}$ induced by the universal property must therefore be a surjection, for it is a composition of a surjection with an isomorphism, namely the inverse of $(p_{2})_{p}$. The first isomorphism theorem implies $(f^{*}\operatorname{T}N)_{p}$ is a quotient of $(\operatorname{T}M)_{p}$.
Finally, the kernels at each point $p$ can be glued together to a bundle. Just take a trivializing open cover for $\operatorname{T}M$, a trivializing open cover for $f^{*}\operatorname{T}N$, and take pairwise intersections. Thus, $f^{*}\operatorname{T}N$ is a quotient of $\operatorname{T}M$ by this bundle.
If you have not seen the construction of the pullback bundle: At each point $p$, define $(f^{*}\operatorname{T}N)_{p}$ to be $(\operatorname{T}N)_{f(p)}$. These fibres glue together as follows. Choose an open cover $(U_{\alpha})_{\alpha \in A}$ of $N$ by Euclidean neighborhoods. Such an open neighborhood also trivializes the tangent bundle $\operatorname{T}N$. Moreover, for each intersection $U_{\alpha} \cap U_{\beta}$, we have a smooth map $g_{\alpha\beta}\colon U_{\alpha} \cap U_{\beta} \to \operatorname{GL}_{\operatorname{dim} N}(\mathbf{R})$ which specifies the gluing data. Crucially, these maps $g_{\alpha\beta}$ satisfy the cocycle condition. $$ g_{\alpha\beta} \circ g_{\beta\gamma} = g_{\alpha\gamma}. $$ We can pull back the open cover $U_{\alpha}$ along $f$ to an open cover of $M$. The open sets may no longer be Euclidean, but everything else, e.g., the cocycle condition, survives. In particular, we define a bundle on $M$ with fibre $(\operatorname{T}N)_{f(p)}$ at each point $p$. One can show the bundle so constructed satisfies the universal property of the pullback.