Details of gluing sheaves on a cover

Personally, I find it more intuitive to define the glued-up sheaf $\mathcal F$ like this:

For every open set $V \subset X$, we define the group of sections $\mathcal F(V)$ to be a set consisting of all tuples $(s_i)_{i \in I}$, where each $s_i$ is a section in $\mathcal F_i(V \cap U_i)$, and where the $s_i$'s are required to obey the compatibility condition: $$\phi_{ij}(s_i|_{V \cap U_i \cap U_j}) = s_j |_{V \cap U_i \cap U_j} \ \ \ \ \ (\ast)$$ for all $i, j \in I$. The group addition on $\mathcal F(V)$ is the obvious one.

As far as I can tell, the $\mathcal F$ that I defined is guaranteed to be a sheaf, regardless of whether we impose the cocycle condition. It comes with a natural restriction map, making it a presheaf, and it also obeys all the gluing conditions necessary for it to be a sheaf. I don't think the cocycle condition is needed to verify any of these things.

However, it is not enough to prove that our $\mathcal F$ is a sheaf. We also need to satisfy ourselves that the restriction $\mathcal F|_{U_k}$ really is isomorphic to the $\mathcal F_k$ that we started with, for each $k \in I$. It is here that the cocycle condition is required.

It is easy to write down what the isomorphism $\psi : \mathcal F_k \overset{\cong}\to \mathcal F|_{U_k}$ ought to be. Given an open $V \subset U_k$ and given a section $s \in \mathcal F_k$, we would like to define its image under $\psi$ to be $$ \psi(s) = (\phi_{ki}(s|_{V \cap U_i}))_{i \in I}$$ However, we need to be sure that the tuple $(\phi_{ki}(s|_{V \cap U_i}))_{i \in I}$ represents a well-defined element of $\mathcal F(V)$. In particular, we must verify that $(\phi_{ki}(s|_{V \cap U_i}))_{i \in I}$ obeys the condition $(\ast)$, which states that $$ \phi_{ij} \circ \phi_{ki}(s|_{V \cap U_i \cap U_j}) = \phi_{kj}(s|_{V \cap U_i \cap U_j})$$ for any $i, j \in I$. This is true by virtue of the cocycle condition.