We know the tensor-hom adjuction where the functors are $\space \_ \otimes M$(left adjoint) and $Hom(M,\_)$(right adjoint). Now after this I wanted to find a left adjoint functor to the (left exact) functor $Hom(\_ , M)$,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 .
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.
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 $t:a_i\to a$.