The following is a quotation from Atsushi Moriwaki " The Mordell-Faltings theorem" Remark 2.20
We set $X = \mathbb{P}^n$, $Y = \mathbb{P}^m$, and $Z = \mathbb{P}^n \times \mathbb{P}^m$. We denote by $p_1 \colon Z \to X$ and $p_2 \colon Z \to Y$ the natural projections. Using the seesaw theorem, let us show that the homomorphism $\iota \colon \mathbb{Z} \times \mathbb{Z} \to Pic(Z)$ defined by $(a,b) \mapsto O_Z(a,b) : = p_1^*O_X(a) \otimes p_2^*O_Y(b)$ is an isomorphism. Indeed, the injectivity of $\iota$ follows from $\deg(O_Z(a,b)|_{X \times \{ y_0 \} }) = a$, $\deg(O_Z(a,b)|_{ \{ x_0 \} \times Y }) = b$ for any $x_0 \in X(F)$ and $y_0 \in Y(F)$.
My question: What is the definition of $\deg(O_Z(a,b)|_{X \times \{ y_0 \} })$?
A line bundle on $X = \Bbb P^n$ is on the form $L = \mathcal O_X(d)$ for some integer $d \in \Bbb Z$ and by definition $\text{deg}(L) = d$.