What is a good name for the following definition pattern? I am often in the following situation. I have a category $\mathbb C$ and a functor $F$ out of it. Sometimes it happens that there is an object $X$ in $\mathbb C$ such that for any other object $Y$ the map of sets $\operatorname{Hom}_\mathbb C(X,Y)$ to $\operatorname{Hom}(FX,FY)$ is a bijection. What is a good way to say that this is the case (in informal language, not symbols)?
I am tempted to say that in this case $X$ is freely generated by its $F$. Examples:

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*Take the category of schemes and the functor $\mathcal O:Sch\to Ring$ which sends a scheme to its ring of regular functions $\mathcal O(X) = \operatorname{Hom}_{Sch}(X,\mathbb A)$. I would say that $X$ is cofreely generated on its ring of regular functions if and only if for any other scheme $Y$, the morphism between the sets $\operatorname{Hom}_{Sch}(Y,X)$ and $\operatorname{Hom}_{Ring}(\mathcal OX,\mathcal OY)$ is an isomorphism. Those are precisely the affine schemes.


*Take the category of $\mathcal O_X$-modules in the little topos $Sh(X)$ of a scheme. There is a global section functor $\mathcal O_XMod \to \mathcal O(X)Mod$, and I would say that an $\mathcal O_X$-module $\mathcal F$ is freely generated by its global sections if and only if for each other $\mathcal O_X$-module $\mathcal G$ we have that $\operatorname{Hom}_{\mathcal O_XMod}(\mathcal F,\mathcal G)\to \operatorname{Hom}_{\mathcal O(X)Mod}(\mathcal F(X),\mathcal G(X))$ is a bijection. In the case that $X$ is affine, those are precisely the quasicoherent $\mathcal O_X$-modules (if I am not mistaken).
What is a good name for my definition pattern. The trouble is that $X$ is freely generated by $S$ normally means that $X = FS$ for some left adjoint functor $F$, and I do not want to confuse the concepts.
 A: I don't know a name for this condition unless $F$ has a left adjoint, which to avoid confusing myself I will name $H$. Then the condition is equivalent, by the Yoneda lemma, to the condition that the counit $H(F(X)) \to X$ of the adjunction is an isomorphism on $X$; in this case one says that $X$ is a fixed point of the adjunction.
A standard fact about adjunctions is that every adjunction restricts to an equivalence of categories between the fixed points on each side; when applied to the example of the adjunction between schemes and affine schemes this recovers the equivalence of categories between affine schemes and, well, affine schemes. Similarly for quasicoherent $\mathcal{O}_X$-modules on affine schemes, I think.
Note that this condition implies that $\text{Hom}(F(X), F(-))$ preserves limits, and the easiest way for that to happen is that $F$ itself preserves limits, and the easiest way for that to happen is that $F$ has a left adjoint, so the extra condition that $F$ has a left adjoint is pretty mild here and likely to hold in examples where this condition holds anyway.
