Let $C_1, C_2$ be two curves (so $1$-dimensional, proper k-schemes) such that their dualizing/canonical sheaves are the sheaves of Kähler differentials; therefore $\omega_{C_1}^1 = \Omega^1 _{C_1/k}$ (resp. $\omega_{C_2}^1 = \Omega^1 _{C_2/k}$); for example if $C$ and $C'$ are smooth.
Let consider the surface $S:= C_1 \times C_2$. My question is why and how to see that
$$\Omega_{S/k}= pr^*_1(\Omega_{C_1}^1) \oplus pr^*_2(\Omega_{C_2}^1)$$
where $pr_i: S \to C_i$ canonical projections.
Obviously one can this prove locally, so in case that $C_i$ are affine.
How can one in this case intuitively (or on level of rings) imagine the tensor product of pull backs $pr^*_i(\Omega_{C_i}^1)$ of Kähler differentials?