If $X$ is a scheme, then by a representable étale sheaf one means the following: for a scheme $Y\to X$ over $X$, we may consider the presheaf of sets $$U \mapsto \operatorname{Hom}_X (U,Y),$$ and it is actually a sheaf on $X_\text{ét}$.
I've seen several times the claim that every sheaf on $X_\text{ét}$ is a colimit of representable sheaves. Could someone give a reference for that? (I guess it must be proved and used in SGA 4.)
Can we assume that the representable sheaves in question are represented by étale schemes $Y\to X$?
Thank you.
Every presheaf is a colimit of representable presheaves, which are in fact sheaves as the \'etale topology is subcanonical (that is, every representable functor is a sheaf).
Let $\mathscr{F}$ be any sheaf on $X_{et}.$ If $\iota : \mathsf{Sh}(X_{et})\to\mathsf{Psh}(X_{et})$ is the forgetful functor including sheaves on $X_{et}$ into presheaves on $X_{et},$ we may write by the above $\iota(\mathscr{F}) \cong \operatorname{colim}h_U.$ Now, recall that we have an adjoint pair $$ (-)^{++}:\mathsf{Psh}(X_{et})\leftrightarrows\mathsf{Sh}(X_{et}): \iota $$ ($(-)^{++}$ denotes sheafification) and that left adjoints preserve colimits and right adjoints preserve limits. Thus, it follows that (because there is a natural isomorphism $(-)^{++}\circ\iota\to\operatorname{id}_{\mathsf{Sh}(X_{et})}$) $$ \mathscr{F}\cong (\iota(\mathscr{F}))^{++}\cong(\operatorname{colim}h_U)^{++}\cong\operatorname{colim} (h_U)^{++}\cong\operatorname{colim} h_U. $$