left adjointable functors When a functor $F$ is left adjoint to some functor $G$, then one usually says that "$F$ is a left adjoint". Is this grammatically correct? Wouldn't it be more accurate to say that "$F$ is right adjointable"? Similarly, one already speaks of (left or right) "dualizable objects" in a monoidal category. Notice that both notions, adjointness and duality, can be unified via $2$-categories. Googling yields "left adjointable" only in the context of operator algebras.
In German, the phrase "$F$ is ein Linksadjungierter" looks even more awkward to me; "$F$ ist rechtsadjungierbar" is much better.
Other suggestions are also welcome.
 A: In principle, "$F$ is a left adjoint" is fine because the adjunction is determined uniquely up to unique isomorphism by $F$. This seems to contradict your intuition that "$F$ is a left adjoint" sounds wrong, but it's perfectly consistent with standard usage such as talking about "the" limit of $G$: this sort of usage works fine except in contexts where it's important to distinguish between uniquely isomorphic objects. In such contexts, we can say things like "a limit of $G$", or "$F$ is a functor which admits a right adjoint". It's okay that these expressions are awkward because context doesn't often demand that we resort to them. In principle, we could do everything in Homotopy Type Theory, where uniquely isomorphic things are literally equal, and these strange contexts couldn't even come up.
So the real question is why we say "dualizable", even in contexts where we don't need to distinguish between uniquely isomorphic things. I suspect it's because often in a context where it makes sense to ask whether an object is dualizable, there are actually two notions at duality at play, one in the adjoint sense and one in another sense. For example, in vector spaces, there's a non-adjoint sense of "dual" given by $\mathrm{Hom}(-,k)$. We say "dualizable" to force the interpretation of the word into the adjoint sense. This just happens to work because in the non-adjoint sense, every vector space $V$ has a "dual" $\mathrm{Hom}(V,k)$ so it wouldn't make sense to ask whether it's "dualizable".
