Under what conditions can one glue together local diffeomorphisms

Question

I know that, given an open cover $\{U_\alpha\}$ of a manifold $M$ and a family of smooth maps $\,f_\alpha: U_\alpha \to N$ that agree on overlaps, it is possible to construct a smooth $f:M \to N$ that restricts to $f_\alpha$ on each $U_\alpha$.

My question is, are there conditions under which you can do the same thing with a family of local diffeomorphisms? I have a family of such maps defined on open subsets of some $M$ that agree on overlaps, and I would like to glue them together on some larger open set and have the extended map remain a local diffeomorphism. Is such a thing possible?

Thanks,

EDIT: To be more clear, I suppose I mean conditions on which the extended map is a diffeomorphism on the entire larger open set. It is obviously still a local diffeomorphism after extension.

Answer 1

Lemma: A local diffeomorphism is a diffeomorphism if and only if it is bijective

E.g. see this mse question.

So, your question simply reduces to checking whether the total function $f : M \to N$ is bijective.

If it's awkward to tackle the question directly, note that the condition that $f$ is injective is equivalent to the condition that whenever $f_\alpha(x) = f_\beta(y)$, then you also have $x=y$.