= Solution
Let $\pi_i:\mathbb C^{g_i}\to X_i$ be the quotient <covering maps>. Since the source universal cover is <simply connected>, the <lifting criterion for a covering space> gives a lift $F:\mathbb C^{g_1}\to\mathbb C^{g_2}$ of $f\circ\pi_1$. The lift is a <holomorphic map> because $\pi_2$ is locally a <biholomorphism>.
For every $\lambda\in\Lambda_1$, the difference $F(z+\lambda)-F(z)$ belongs to the discrete <Euclidean lattice> $\Lambda_2$. It is a continuous function of $z$ on a <connected> space, hence is constant. Differentiating shows that each coefficient of $DF$ is $\Lambda_1$-periodic. Such a coefficient is bounded on a compact fundamental parallelepiped, and periodicity bounds it on all of $\mathbb C^{g_1}$. The several-variable <Liouville theorem>, obtained by applying the one-variable theorem on coordinate lines, makes every coefficient constant.
Consequently $DF=A$ for a constant <complex-linear map> $A$ and $F(z)=Az+x$. The period differences become $A\lambda\in\Lambda_2$. Thus the <affine lift of a holomorphic map between complex tori> gives
$$
\boxed{f(z+\Lambda_1)=Az+x+\Lambda_2,\qquad A\Lambda_1\subseteq\Lambda_2.}
$$
\b[Every holomorphic map between complex tori is a homomorphism followed by a translation.] The translation vector $x=F(0)$ is determined modulo $\Lambda_2$, while $A$ is the derivative of any lift and is unaffected by that ambiguity.
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