If I understand correctly, the following is Theorem 1.32 of Warner's Foundations of Differentiable Manifolds book.
Proposition. Let $X\to Y$ be smooth and $\iota :A\to Y$ be an immersed submanifold. Suppose moreover $f(X)\subset\iota (A)$. By injectivity of $\iota$ there's a unique set function $f_0$ such that $\iota\circ f_0=f$.
- $\iota$ is a topological embedding $\implies f_0$ continuous.
- $f_0$ continuous $\implies f_0$ smooth.
In 1.33(a) we have the following assertion.
Let $M$ be a differentiable manifold and $A\subset M$. Fix a topology on $A$. Then there is at most one differentiable structure on $A$ such that $\iota:A\hookrightarrow M$ is an injective immersion.
The hint is to apply Theorem 1.32, but I just don't see it. How to prove this result?