Let $k$ be a field of arbitrary characteristic and let $X$ and $Y$ be projective varieties over $k$. I have come across the formula $\operatorname{Pic}^0(X) \times \operatorname{Pic}^0(Y) \cong \operatorname{Pic}^0(X \times Y)$ several times but never with a proof nor a reference to a proof nor the precise conditions needed on $X$, $Y$ and $k$. Does anyone have a reference or a short proof of this fact? Let $\pi_X:X\times Y \to X$ and $\pi_Y:X\times Y \to Y$ be the projection maps. Is the isomorphism given by $(\mathcal{L}, \mathcal{M})\mapsto \pi_X^*(\mathcal{L}) \otimes \pi_Y^*(\mathcal{M})$, or something more exotic?
NB: I'm happy to leave it up to the answerer to choose the most appropriate definition for $\operatorname{Pic}^0(X)$, though I am thinking about it as the kernel of the natural map $\operatorname{Pic}(X)\to \operatorname{NS}(X)$ from the Picard group to the Néron-Severi group. The definition as the connected component of the scheme $\operatorname{Pic}(X)$ might be profitable.