One can use instead of the spectrum an ${{\bf R}}$-filtration
$Hom_{{\cal C}(Y)}(E_{1},E_{2})^{\leq s}$ on the space of morphisms. It comes from the filtration on the stalks of sheaves of morphisms $\underline{Hom}_{{\cal O}_{Y}}(E_{1},E_{2})\widehat{\otimes}\widehat{\Omega}_{Y}^{\ast}$ (completed tensor product) defined by the condition $\{-\lambda_{j,i_{1}...i_{n}}+\sum_{1\leq k\leq n}i_{k}y_{k}+f_{1}(y)-f_{2}(y)\}\leq s$. It is easy to see that $\alpha$ belongs to $Hom_{{\cal C}(Y)}(E_{1},E_{2})^{\leq s}$ iff for all $y\in Y$ one has $Sp_{y}(\alpha)\subset(-\infty,s]$.