I am trying to understand the proof of the following theorem that can be found in Affine Group Schemes Chapter IV, Appendix.
Particularly, I do not understand why if $F'$ is the restriction of the functor $F$ to $k'$ algebras (via $k\to k'$) it is equipped with a descent datum. I have done the following
This isomorphism is probably the descent datum I am looking for. However I am having problems trying to verify the cocycle conditions that can be found here in Definition 35.3.1.
I thought about using the different maps $p_{ij}\colon k'\otimes_k k' \to k'\otimes_k k'\otimes_k k'$ that map an element $a\otimes b$ to the corresponding $ij$ entries. Then induce via $\varphi\colon A'\otimes_k k'\to k'\otimes_k A'$ (the isomorphism we found by Yoneda lemma) the maps that lead to the cocyle condition. However, I am not able to see why the diagram in the definition for descent datum is commutative for this particular isomorphism.