Let consider the scheme $S:= \mathbb{P}^1 \times \mathbb{P}^1$ and denote the canonical projection morphism by $pr_i: S \to \mathbb{P}^1$ for $i = 1,2$.
Define invertible sheaf $\mathcal{L}:= pr_1 ^*(\mathcal{O}_{\mathbb{P}^1}(1))$ on $S$.
Let $a \in \mathbb{P}^1$ such that $pr_i^{-1}(a) \cong \mathbb{P}^1$.
My question is why does for the restriction of $ \mathcal{L}$ to $pr_2^{-1}(a) $ holds
$$\mathcal{L} \vert_{pr_2^{-1}(a)} = \mathcal{O}_{\mathbb{P}^1}(1)$$
Any $a \in \Bbb P^1$ verify $pr_i^{-1}(a) \cong \Bbb P^1$ (if at least you consider only closed points). Anyway, if we rephrase everything in terms of divisors, notice that $$(\{a'\} \times \Bbb P^1) \cap (\Bbb P^1 \times \{a\}) = \{a'\} \times \{a\}$$
which means exactly that $L_{|pr_2^{-1}(a)} \cong \mathcal O_{\Bbb P^1}(1)$.
This is because if $D \subset X$ is a divisor and $Y \subset X$ is a variety, then $D$ corresponds to a line bundle $L$ and $Y \cdot D$ (intersection product) is a divisor on $Y$, corresponding to $L_{|Y}$.
Also I should say that usually what is called the tautological sheaf is $\mathcal O_{\Bbb P^1}(-1)$.