Let $(X, L)$ and $(Y, M)$ be two polarized abelian varieties . 


According to Birkenhake C. and Lange H. in Complex Abelian Varieties a homomorphism of polarized abelian varieties $f:(Y, M)\longrightarrow (X, L)$ is a homomorphism of complex tori $f:Y\longrightarrow X$ such that $f^{\*}c\_1(L) = c\_1(M)$. 


Question: It's true that $f^\*c\_1(L)=c\_1(f^\*L)$?



Pull-back of polarization
Yes! This is sometime called naturality of Chern classes. You can find it in many books, for instance Complex Geometry - An Introduction | Daniel Huybrechts, or Differential forms in algebraic topology by Bott and Tu.


