Convention:
Before I start, please note that it is not going to be sufficient to assume that the topology is subcanonical or that the site $C$ has finite limits, since the application I have in mind does not satisfy either of these two criteria. Therefore, we take our definition of a topology to be the one in terms of covering sieves (a function $J$ assigning to each object $U$ a family of subfunctors $S\subseteq h_U$ satisfying conditions). Each sieve has a canonical "cover" attached to it, $\{U_i\to U\}_{i\in I}$ (although because our case is somewhat pathological, the resulting system of covers does not meet the requirements for being a Grothendieck pretopology (the requirements are not meaningful in this setting because our underlying category does not have pullbacks)). We will make use of these covers in a construction, but it is to be understood that these arise from an actual Grothendieck topology.
Let $(C,\tau)$ be a Grothendieck site. It's well-known that the double-plus construction $(-)^{++}$, which is actually the composite $((-)^+)^+$ is the left adjoint to the inclusion functor $Sh(C,\tau)\hookrightarrow Psh(C)$. Recall that $(-)^+$ replaces the functor $F(-)$ by the functor $H_\tau^0(-,F)$, or the $0^{th}$ sheaf of Čech cohomology, which can be constructed as the colimit $H^0_{\tau}(U,F)=L_\tau F(U):=Colim_{S\in J(U)^{op}} Hom(S,F)$.
Up until yesterday or so, I believed that the functor $(-)^+:Psh(C)\to SepPsh(C)$ is left adjoint to the inclusion $\iota:SepPsh(C,\tau)\hookrightarrow Psh(C)$, but I feel very inclined to doubt it now, since by a computation I did with a toy example, it doesn't seem to have the right universal property (I'm unsure, because the one place that I've found that addresses the issue says that it is).
However, when I was looking back through Vistoli's notes in the section on sheafification, I noticed that his construction of the sheafification is actually as $((-)^{sep})^+$, where $(-)^{sep}$ is defined by quotienting out by the following equivalence relation: $R(U)\subset F(U)\times F(U)$ where a pair $(a,b)\in F(U)\times F(U)$ is in $R(U)$ (that is to say, $a\sim b$) if there exists a cover $\{U_i\to U\}_{i\in I}$ such that $a|_{U_i}=b|_{U_i}$ for each $i\in I$.
First thing: It's clear that the relation is reflexive and symmetric, but is it also clear that it's transitive? If it's not, does Vistoli's construction work if we saturate the relation to an equivalence relation?
Second thing: Is the $(-)^+$ functor actually the left-adjoint of the inclusion $\iota$? If not, is $(-)^{sep}$? If the left adjoint is indeed $(-)^{sep}$, does it preserve monomorphisms (this is not to show that it's left-exact, since that will definitely fail (since separated presheaves do not form a topos). I have something totally different in mind unrelated to the rest of this question).
The toy example I mentioned above is the one to keep in mind, so I'll mention it here:
Consider the category $[1]$, which is the category with two objects $0$ and $1$ with one nonidentity morphism $0\to 1$. Let $S=h_0\subseteq h_1$ be the only nonidentity covering sieve of $1$. Then separated presheaves on this site are exactly the functors $F:[1]^{op}\to Set$ such that the restriction map $res_{0}:F(1)\to F(0)$ is injective. Given a presheaf $F$, $F^{sep}$ is given by the quotient where $F^{sep}(0)=F(0)$ and $F^{sep}(1)$ is the quotient of $F(1)$ by the equivalence relation noted above.
Notice also that sheaves for this topology are the presheaves such that $F(1)\to F(0)$ is a bijection.
We can easily verify the universal property of Vistoli's construction, since given a morphism $F\to G$ where $G$ is separated, we see that it obviously factors uniquely through the quotient. The $(-)^+$ construction does not seem to have this property in any obvious way.
