Homomorphisms of Tori Correspond to Lattice Maps
Let \(E/k\) be a finite Galois extension with group \(\Gamma = \operatorname{Gal}(E/k)\), and let \(T, T'\) be tori over \(k\) split by \(E\). Then the character-lattice functor gives a bijection \[ \operatorname{Hom}_{k\text{-grp}}(T,\, T') \;\;\xrightarrow{\;\sim\;}\;\; \operatorname{Hom}_{\Gamma}\bigl(X^*(T'),\, X^*(T)\bigr), \qquad f \longmapsto X^*(f) = (\chi \mapsto \chi \circ f_E), \] where the right-hand side is the group of \(\Gamma\)-equivariant homomorphisms of the character lattices (Character and Cocharacter Lattices). The bijection is contravariant and compatible with composition and identities.
Proof
Step 1: descent of homs. \(k\)-homomorphisms \(T \to T'\) correspond to \(k\)-Hopf algebra maps \(k[T'] \to k[T]\), and by the hom part of Galois Descent for Hopf Algebras these correspond to \(\Gamma\)-equivariant \(E\)-Hopf maps \(E[T'_E] \to E[T_E]\), i.e. \[ \operatorname{Hom}_{k\text{-grp}}(T, T') \;\cong\; \operatorname{Hom}_{E\text{-grp}}(T_E,\, T'_E)^{\Gamma}, \] where \(\Gamma\) acts on an \(E\)-homomorphism by conjugating with the semilinear actions on both coordinate algebras.
Step 2: the split case. Over \(E\) both tori are split, so by Cartier duality for diagonalizable groups (Diagonalizable Groups Are Dual to Character Groups) \[ \operatorname{Hom}_{E\text{-grp}}(T_E,\, T'_E) \;\cong\; \operatorname{Hom}_{\mathbb{Z}}\bigl(X^*(T'),\, X^*(T)\bigr), \] contravariantly: an \(E\)-homomorphism of split tori is determined by its effect on characters, and every lattice map arises (on group algebras, \(e^{m'} \mapsto e^{\phi(m')}\)).
Step 3: matching the actions. The \(\Gamma\)-action of Step 1 on \(E\)-homomorphisms corresponds under Step 2 to the conjugation action on lattice maps, \(\phi \mapsto \sigma_\gamma \circ \phi \circ \sigma'^{-1}_\gamma\): this is the computation that the semilinear action permutes the group-like bases by the lattice actions (Characters Are Group-Like Elements transported as in The Twisted Form of a Galois Lattice property (3)). Fixed points of the conjugation action are exactly the \(\Gamma\)-equivariant lattice maps. Combining the three displays gives the bijection; functoriality holds because each step is compatible with composition. \(\square\)