Yes. Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism from $Y$ to $G$.
In fact, if you can find an initial morphism from $G$ to each $Y$, then one can prove that the association $X\mapsto T_X$ extends to act on morphisms in a unique, natural way to define a functor $F$, which moreover is right adjoint to the original functor $F$. This is elementary, but not trivial, and is explained relatively well here.
As an example to illustrate how helpful this can be, think of the inclusion functor $U$ from AbGrp to Grp. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H] \in AbGrp$ satisfying an initial (universal) property, so by the above theorem, we know this association automatically in a unique way to to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.