I hope this question doesn't sound lame, but I have a few problems in understanding the notion of Intersection product.
Clearly, for a (Weil) divisor $D$ we have the first Chern class $c_1(\mathcal{O}(D)) \in H^2(X,\mathbb{R})$, where $\mathcal{O}(D)$ is the corresponding line bundle. Let $f \colon X \to Y$ be a morphism of projective varieties. Let $C$ be a curve in $X$ and $D$ be a divisor on $Y$. I want to compute the intersection product $f^*(\mathcal{O}(D)).C$.
By definition, this is $$ f^*(\mathcal{O}(D)).C = c_1(f^*\mathcal{O}(D)).C = \int_C c_1(f^*\mathcal{O}(D)) = \int_C f^*(c_1(\mathcal{O}(D)) $$.
So far, so good. I am wondering now, whether we are able to identify the right-hand side with $$ \int_{f(C)} c_1(\mathcal{O}(D)) = \mathcal{O}(D).f(C)?$$ Why would this be a correct thought? Or why not?