Isomorphism of schemes and invertible sheaves

Question

I have a question more about terminology than anything else: If $f:X\rightarrow Y$ is an isomorphism of schemes, then what does it mean to say "the invertible sheaf $\mathscr{L}$ on $X$ corresponds to the invertible sheaf $\mathscr{M}$ on $Y$?" Naturally, it makes sense to assume that this statement means $f_{\ast}\mathscr{L}=\mathscr{M}$ or $\mathscr{L}=f^{\ast}\mathscr{M}$, but I wasn't sure...