Yes, and it is easy to check!  In fact, the following theorem makes precise why "this looks like it might work" is so often successful:

<b>Theorem:</b> 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][1] $\phi_Y:Y\to G(T_Y)$ from $Y$ to $G$.   Moreover, once you find such an initial morphism from each $Y$ to $G$, the association $Y\mapsto T_Y$ extends in a unique way (really unique, i.e. not just up to natural isomorphism) to act on morphisms defining a functor $F$, which *moreover* is left adjoint to the original functor $G$.  

This is not hard to prove, but not trivial, and is explained relatively well [here.][2]  Once you know this, you can really take confidence in "follow your nose"-style adjoint construction.

As an example of how this can be useful, 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.  But then by the above theorem, we know this association automatically extends in a unique way to to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.

This same trick expedites the construction of adjoints in pretty much any situation you can think of.  I think it's really worthwhile to sift through the three different characterizations of adjoints given on [Wikipedia][3].


  [1]: http://en.wikipedia.org/wiki/Universal_morphism
  [2]: http://en.wikipedia.org/wiki/Universal_morphism#Relation_to_adjoint_functors
  [3]: http://en.wikipedia.org/wiki/Adjoint_functors