Yes, and it is easy to check. The following result makes precise why "this looks like it might work" is so often successful:
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 $\phi_Y:Y\to G(I_Y)$ 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 $G$.
This is well-known and easy to prove (well, depending on who you ask), but is non-trivial and involves many steps, which are explained relatively well here. (Essentially one is recovering the entire adjoint situation from just one functor and a unit transformation.)
Once you know it, you can really take confidence in "follow your nose"-style adjoint construction. It doesn't involve having an "initial guess" for the left adjoint (as a functor), 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.
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 $AbGrp$ 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.
Edit: 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. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.