The Stacks project

To finish the proof don't we need to show also that $Y\mapsto colim_i F(Y_i)$ satisfies the sheaf axiom? I understand that the presheaf pullback $F'=\epsilon_p(F)$ of $F$ satisfies the conclusion $F'(Y)=colim_{i} F(Y_i)$ (since coequalisers in presheaves are computed pointwise) but it's not obvious to me why should the sheafification of $F'$ satisfy this equality too.

No because we are directly computing the values of the sheaves! I tried to clarify this in the following edit.

if F is a sheaf on $X_{\text{étale}}$ then shouldn't the $Y_i$ be objects of $X_{\text{étale}}$ if we are to evaluate $F(Y_i)$?

I think I understand what the statement should be (as needed to prove 61.19.4):

Let $X$ be a scheme. Let $Y=\lim Y_i$ be the limit of a directed inverse system of quasi-compact and quasi-separated objects of $X_{\text{pro-étale}}$ with affine transition morphisms. For any sheaf $F$ on $X_{\text{étale}}$ we have $\epsilon^{-1}F(Y)=\operatorname{colim}_i\epsilon^{-1}F(Y_i)$. If the $Y_i$ are moreover objects of $X_{\text{étale}}$, we have $\epsilon^{-1}F(Y)=\operatorname{colim}_i F(Y_i)$.

The proof is the same and uses that $\epsilon^{-1}$ commutes with colimits and colimits commute with colimits; i.e. if $C$ denotes $X_{\text{étale}}$, once we have $\epsilon^{-1}h_U(Y)=\operatorname{colim}\epsilon^{-1}h_U(Y_i)$, we write

$\epsilon^{-1}F(Y)=(\epsilon^{-1}\operatorname{colim}_{C/F}h_U)(Y)=\operatorname{colim}_{C/F}\epsilon^{-1}h_U(Y)=\operatorname{colim}_{C/F}\operatorname{colim}_i\epsilon^{-1}h_U(Y_i)$ $=\varinjlim_i\operatorname{colim}_{C/F}\epsilon^{-1}h_U(Y_i)=\operatorname{colim}_i\epsilon^{-1}F(Y_i).$

When the $Y_i$ are in $X_{\text{étale}}$, we have $\epsilon^{-1}h_U(Y)=\operatorname{colim}h_U(Y_i)$ and the above simplifies.

Thanks very much since this was quite a mess. But I could not make your fix work because we don't know that taking sections over Y commutes with the colimits in question (they aren't filtered). In fact, my first fix was wrong for this reason. I fixed it instead by characterizing the pullback sheaf first (and this should have been done a long time ago --- I was just lazy) and then using the same result for etale sheaves which already was in the stacks project. Corresponding commits on github are: wrong fix and correct fix.