Let us do the case of an affine scheme $X$ first. This is easy. If $M$ is an $O_X$-module, we define $\tilde{M}$ as the quasicoherent $O_X$-module defined by the global sections $M(X)$. Notice that the restriction gives the canonical map $\tilde{M}->M$.
As step 2, we extend to the case of quasiaffine $X$. Quasiaffinity means that $X-> Spec O_X(X)$ is an open embedding. Equivalently, all its quasicoherent modules are generated by global section. Hence, the coherator functor $M |-> \tilde{M}$ is defined in the same way. BTW, it is clear how the coherator works on maps too.
Finally, we have everything ready as a general scheme admits an affine open cover $X=\cup_i U_i$. The quasiaffine case is useful as double and triple intersection $U_{i,j}$, $U_{i,j,k}$ are all quasiaffine. Given an $O_X$-module $M$, we use its open pieces $M_i=M|_{U_i}$ and gluing maps $\phi_{i,j}$ from ${M_i}|_{U(i,j)}$ to ${M_j}|{U(i,j)}$ ({M_j}|_{U(i,j)}$,
where $U(i,j)=U{i,j}$ U(i,j)=U_{i,j}$ (because tex translator does not like me) is finding it difficult to comprehend it too without each formula starting in a new line), that satisfy the cocycle conditions on triple intersection.
Now the coherator is glued from open pieces $\tilde{M}i$ \tilde{M}_i$
using isomorphisms $\tilde{\phi}{i,j}$ \tilde{\phi}_{i,j}$ which inherit the cocycle conditions. And That's All Folks!
