I hope my question is not too basic here.

I have a very specific setting and I am trying to not use "big theorems" to prove a well known fact algebraically. I hope someone can give me a hint.

Let $J/k$ be the Jacobian of a hyperelliptic curve $H$ of genus $2$ with one rational point at infinity $\infty$. Consider $\iota:H\to J$ given by the usual embedding $\iota(P)=[P-\infty]$. Let $\Theta:=\text{Im}(\iota)\in\text{Div}(J)$ be the theta divisor of $J$.

I need to use that $\deg\iota^*\Theta=2$, that is, $\iota^*\Theta=2\infty$. Poincare's formula helps as this is equivalent to calculate the self intersection $\Theta\bullet\Theta$, but I think a simpler argument could help using that $J$ is the jacobian of a hyperelliptic curve $H$ (which has a symmetric theta divisor isomorphic to $H$).

I want to calculate this special case by hand but I have some problems understanding the pullback of $\Theta$ under $\iota$. (In fact I will appreciate a definition of pullback of morphisms of varieties with different dimensions using Weil Divisors, a lot is documented, but just for curves or in general for line bundles)

I saw a proof in the book $\textit{Diophantine geometry - An Introduction}$ by Silverman and Hindry, theorem A.8.2.1 for general jacobians and non-symmetric Theta divisors but I think in this genus $2$ case it can be simpler.

Also there is this question Pullback of theta divisor Which is also very general.

I am working over $\overline{\mathbb{F}_q}$, and I am trying not to abuse of the $\mathbb{C}$ proofs and then using Lefschetz principle.