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] $\phi_Y:Y\to G(I_Y)$ from $Y$ to $G$.   Moreover, once you find such an initial morphism from each $Y$ to $G$, the association $Y\mapsto I_Y$ extends in a unique way to act on morphisms defining a functor $F: D\to C$, which *moreover* is left adjoint to the original functor $G$.

It is not that hard to prove, but non-trivial and involves many steps, which are explained relatively well [here.][2]  

Once you know this, you can really take confidence in "follow your nose"-style adjoint construction.  Notice that:

<b>(1)</b> This theorem does not involve having an "initial guess" for the left adjoint, but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms (really unique, not just up to natural isomorphism).  

<b>(2)</b> Its conditions are *asymmetric*, but conclusion and the definition of adjoints (using hom-sets) is *symmetric*, which means there really is something to prove here.

<b>As an example</b> 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 $AbGrp$ with a map $H\to Ab(H)$ satisfying an initial (universal) property.  But then by the above theorem, we can automatically extend this association in a unique way 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][2].

<b>Edit:</b> Sometimes this theorem is used as an alternative definition for adjoint functors in terms of universal morphisms. However you look at it, the real utility is knowing that this "weak" , and in fact asymetric, condition actually implies the "stronger", symmetric definitions of adjoints via hom-sets or units/counits.  

For example, it follows further from this theorem that the functor $G$ actually defines provides you with a family of *terminal* morphisms to the functor $F$, which is now many logical steps removed from the assumption that *initial* morphisms existed from $G$.

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