Consider a morphism $f:X\to Y$ between two (quasi-projective, say) algebraic varieties. Let $X_1,\dots,X_n$ be the irreducible components of $X$. Suppose that $f$ is a bijection, that $f(X_1),\dots,f(X_n)$ are the irreducible components of $Y$ and that, furthermore, $f:X_i\to f(X_i)$ is an isomorphism for every $i$. I know that, in general, this does not imply that $X$ and $Y$ are isomorphic. My question is under which reasonable additional conditions $f$ must indeed be an isomorphism.
In case it helps, in my setting $X$ and $Y$ are projective and their components are normal. (Could these conditions already be sufficient?)