If we have the family of curves $f:X\rightarrow Y$, we consider a fiber $C=f^{-1}(y)$ and its tangent bundle $T_{C}$. We know that $df: f^{*}{T_{Y}}_{|C}\cong N_{C}.$ In fact both of these budnles are trivial and isomorphic to $C\times T_{y}$(Voisin pg.223). Can we give explicitely this isomorphism, i.e if we take some vector from $N_{C}$ how do we see it as an element of $f^{*}{T_{Y}}_{|C}?$

If we have the family of complex curves $f:X\rightarrow Y$, over a complex smooth curve $Y$ , we consider a fiber $C=f^{-1}(y)$ and its tangent bundle $T_{C}$. We know that $df: f^{*}{T_{Y}}_{|C}\cong N_{C}.$ In fact both of these budnles are trivial and isomorphic to $C\times T_{y}$(Voisin pg.223). Can we give explicitely this isomorphism, i.e if we take some vector from $N_{C}$ how do we see it as an element of $f^{*}{T_{Y}}_{|C}?$