Equivalent definitions of Verma modules This is a rather basic question. I was reading some notes on geometric representation theory by Gaitsgory and his defition of Verma module is the following: Let $ \lambda $ be a weight of $ \mathfrak{g}$. The Verma module $M_\lambda \in \mathfrak{g} - mod$ is defined such that for any object $ M \in \mathfrak{g} -mod$ we have : $Hom_\mathfrak{g} (M_\lambda , M) = Hom_\mathfrak{b} (\mathbb{C}^\lambda , M)$. Where $\mathfrak{b}$ is the borel subalgebra and $\mathbb{C}^\lambda $ is the one dimensional $\mathfrak{b}$ module.
The definition of Verma module that I am used to is that we define $M_\lambda := U(\mathfrak{g}) \otimes_{U ( \mathfrak{b} )} \mathbb{C}^\lambda$. 
Gaitsgory says that his defintion implies this one but gives no proof and I cant see how it should be. My first thought was that maybe I should play around with some adjoint functors but I don't really see how to proceed. An answer would be nice but maybe its a lot to write so I would love even a hint to get started. Thanks.
 A: The two definitions are seen to be equivalent after considering the universal construction of the tensor product.
Starting from the tensor product definition, we see the Verma module is the most general $U(\mathfrak{b})$-bilinear map that projects down to $U(\mathfrak{g})$ and $\mathbb{C}^{\lambda}$. This is equivalent to forcing the $U(\mathfrak{g})$-linear morphisms of $M_{\lambda}$ to coincide with the $U(\mathfrak{b})$-linear morphisms of $\mathbb{C}^{\lambda}$, which is Gaitsgory's definition.
By the way, I feel it is important to state the action of $\mathfrak{b}$ on $\mathbb{C}^{\lambda}$ because this is the concrete part of the definition.
A: Let me add on to pre-kidney's answer, which to me is just explaining the tensor-hom adjunction by words; algebraically, it can be expressed in the following lines.  (Note that I identify $\mathfrak{g}-mod$ with $U(\mathfrak{g})-mod$, and consider Hom-spaces in the later category)
Let $M(\lambda):= U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\mathbb{C}_\lambda$, we have
$$ \begin{array}{rcl}
Hom_{U(\mathfrak{g})}(M(\lambda),M) &=& Hom_{U(\mathfrak{g})}(U(\mathfrak{g})\otimes_{U(\mathfrak{b})}\mathbb{C}_\lambda,M)\\ &=& Hom_{U(\mathfrak{b})}(\mathbb{C}_\lambda, Hom_{U(\mathfrak{g})}(U(\mathfrak{g}),M))
\\ &=& Hom_{U(\mathfrak{b})}(\mathbb{C}_\lambda, M)
\end{array}
$$
Conversely, in Gaitsgory's definition, the equality has a hidden "restrict $M$ to a $U(\mathfrak{b})$-module" on the right hand side, which is algebraically given by $Hom_{U(\mathfrak{g})}(U(\mathfrak{g}),M)$ (since restricting is the adjoint of inducing), so reading the above three line backwards shows that $M_\lambda$ is indeed $M(\lambda)$.
