I was wondering if there is some relation between this two notions, in general or in some particular context.
I call a functor $F:A\rightarrow B$ realizable if $\forall y\in Ob(B)$ we can find an element $x\in A$ such that $F(x)=y$. For instance, if I am not wrong, the fundemantal group is such a functor.
On the other hand a functor is conservative it it reflects isomorphisms, hence we can say that $f\in Hom(A)$ isomorphism $\Leftrightarrow$ $F(f)$ isomorphism. For instance the functor which to any affine variety assigns its coordinate ring algebra $\mathcal{O}(X)$, is conservative.
In some sense I would intuitively expect that the two notions should not happen together easily since their combination is quite strong.(Much as in the same way as compactness and Hausdorfness are "inversely" related feature for a topological space). In the example we know that sadly the fundamental group is in general not conservative and that $\mathcal{O}(\cdot)$ is realizable only if we restrict to finetely generated algbras with non nihilpotent elements. (Much as in the same way as compactness and Hausdorfness are "inversely" related feature for a topological space). Thanks for any information.