In his paper "Complex analytic connections in fibre bundles" Atiyah defines his famous Atiyah class $\operatorname{at}(E) \in H^1(X,\operatorname{End}(E) \otimes \Omega^1_X)$ for a vector bundle $E$ on a complex manifold $X$ and he shows that the Chern classes of $E$ can be derived from $\operatorname{at}(E)$. But he also gives a definition for the Atiyah class of an arbitrary coherent sheaf $\mathcal F$ on $X$ which is an element $\operatorname{at}(\mathcal F) \in \operatorname{Ext}_X^1(\mathcal F, \mathcal F \otimes \Omega^1_X )$.
Now, for Chern classes it is well known that they behave functorial with respect to maps $f \colon X \to Y$, i.e. $$ c_k(f^* E ) = f^* c_k(E) $$ for any vector bundle $E$ over $Y$. The comparison of the Atiyah-class with Chern classes suggests that such functoriality should also hold for the Atiyah-class of a vector bundle.
But how is it with the Atiyah-class of a coherent sheaf? Say $f \colon X \to Y$ is a map of schemes/analytic spaces and $\mathcal F$ is a coherent sheaf on $Y$.
Question 1: How can we even define the pullback $$ f^* \colon \operatorname{Ext}^1_Y(\mathcal F, \mathcal F \otimes \Omega^1_Y) \to \operatorname{Ext}^1_X(f^* \mathcal F, f^* \mathcal F \otimes \Omega^1_X)? $$
Question 2: With an appropriate definition of the above, do we indeed get $$ \operatorname{at}(f^*\mathcal F) = f^*(\operatorname{at}(\mathcal F ))? $$
Edit: I see that Question 1 has a straightforward answer in the category of sheaves of Abelian groups. There, the pullback functor $f^{-1}$ is exact. A short exact sequence $$ 0 \to \mathcal F \otimes \Omega^1_Y \to E \to \mathcal F \to 0 $$ is pulled back to a short exact sequence of sheaves of Abelian groups on $X$. Using Yoneda's definition of Ext, we get a notion of pullback. However, in the category of (quasi-)coherent sheaves, the pullback functor $$ f^* \mathcal F := f^{-1} \mathcal F \otimes_{f^{-1} \mathcal O_Y} \mathcal O_X $$ is not an exact functor anymore. So what can we do to remedy for that?