Left adjoint to Hom(_,M)

Question

We know the tensor-hom adjuction where the functors are $ \space \_ \otimes M $(left adjoint) and $\operatorname{Hom}(M,\_)$(right adjoint). Now after this I wanted to find a left adjoint functor to the (left exact) functor $\operatorname{Hom}(\_ , M)$,if any,but failed ! I would like to know whether it has a left adjoint and also if there is always a left adjoint(right adjoint) to a left exact(right exact) functor . Also when defining adjoint pair whether we always have to work in locally small categories(as we have to talk about bijections in the category of sets) ? It seems that WIKI does not say anything about locally small category when discussing adjoint pair ! I know only a bit of category theory .Any help is appreciated .

Answer 1

It is well known that a right adjoint preserves all limits that exist in its domain. You are looking for a converse.
Actually, under some conditions, the converse holds. This is known as The Freyd Adjoint Functor Theorem (see Mac Lane, Categories for the working mathematician, chapter V, section 6, Theorem 2).

Freyd Adjoint Functor Theorem. Given a small-complete category $A$ with small hom-sets, a functor $G:A\to X$ has a left adjoint if and only if it preserves all small limits and it satisfies the following Solution Set Condition. For each object $x\in X$ there exists a small set $I$ and an $I$-indexed family of arrows $f_i:x\to Ga_i$ such that every arrow $h:x\to Ga$ can be written as a composite $h=Gt\circ f_i$ for some index $i$ and some arrow $t:a_i\to a$.

Answer 2

If your category is braided, i.e. you have a natural isomorphism $\tau_{AB} : A\otimes B \to B \otimes A$, then you have that $\text{Hom}(\_, M)^{op} \dashv \text{Hom}(\_, M) : \mathcal{C}^{op} \to \mathcal{C}$ assuming $\text{Hom}(\_,M)$ exists in the first place. That is, $\text{Hom}(\_, M)$ is adjoint to itself on the right. The unit-counit definition of an adjunction immediately generalizes to an arbitrary 2-category.